---
title: "Think Again: 100 Puzzles for the Maths and Science Everyone Should Understand"
url: https://evidencepress.org/think-again/
isbn: 9798163160273
license: CC0-1.0 (this source list; the book itself is all rights reserved)
---

# Think Again: 100 Puzzles for the Maths and Science Everyone Should Understand

*Think Again* is a book of one hundred puzzles in which the answer that seems obvious turns out to be wrong. Each puzzle sits on a right-hand page with three possible answers. Overleaf come the right answer and why the tempting ones fail, the idea in a single sentence, an everyday example of where it matters, and where the idea came from and who first worked it out.

The puzzles run from chance and number, through matter, chemistry and life, to the Earth and the sky; the last section asks how we know any of it. No special knowledge is needed. Most puzzles take a minute to read and longer to settle.

ISBN 979-8-16-316027-3 · Paperback, 6 × 9 in · Evidence Press, 2026. The paperback will be available from Amazon.

## How the facts were checked

Every number in the book was calculated by a script rather than copied from elsewhere, and numbers invented to make an idea clear are labelled as illustrative. Historical claims were checked against the named sources listed below, and a popular story that is better known than it is documented is labelled as a legend in the book and marked † here. The health, body and safety content of 26 puzzles was checked line by line against guidance from the NHS, UKHSA, WHO, CDC, NICE and MHRA, Cochrane reviews and primary papers. A second AI model then reviewed the text adversarially, looking for errors and for anything that could mislead a reader, and the text was corrected where it found a problem.

The book was written with AI, and its checks were made by AI and by scripted calculation. It has not been peer reviewed, and no clinician has read it. It is for general education and is not medical, financial or legal advice; questions about an individual's health belong with a qualified health professional.

## Reading the source list

Each entry pairs a claim made in the book with the source it rests on. The note in small type says how the claim was checked: **Checked** against the source named; **Two or more sources agree**; **Corrected**, where an earlier draft was wrong and the text was changed; **Calculated**, worked out for the book; **Secondary**, where the original document was not seen and the claim rests on a reliable secondary account; **Standard**, a textbook or reference value. † marks a legend or doubtful story, labelled as such in the book. Puzzles with health content also list their **health check**: each health claim, the guidance it was checked against, and whether the text was revised after the check.

Physical constants are CODATA 2018 values. Sources were checked in October 2026. Claims that were checked but left out of the printed book are not listed. A few links no longer resolve; they are kept and marked so that the source can still be traced.

## Sources, puzzle by puzzle

### A. Chance, statistics and evidence (puzzles 1–21)

#### 1. The test says yes

- Bayes was a Presbyterian minister in Tunbridge Wells and died in 1761. Source: MacTutor, "Thomas Bayes": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Biographies/Bayes/) (minister of the Presbyterian chapel, Tunbridge Wells, c. 1733–52; died 17 April 1761). (Checked)
- Price found the essay among Bayes's papers and sent it to the Royal Society, where it was read on 23 December 1763. Source: The essay's own heading, "Read Dec. 23, 1763" (Phil. Trans. 53, 370–418), from the transcription at [york.ac.uk](https://www.york.ac.uk/depts/maths/histstat/essay.htm); MacTutor (above); Stigler, "The True Title of Bayes's Essay", [arXiv:1310.0173](https://arxiv.org/abs/1310.0173). (Two or more sources agree)
- 1978 Harvard study: 60 students and doctors; the commonest answer was 95%; the right answer was about 2%; 11 of 60 gave it. Source: Casscells W, Schoenberger A, Graboys TB, "Interpretation by physicians of clinical laboratory results", *N Engl J Med* 1978;299:999–1001. Figures as summarised by NEJM's CardioExchange blog: [blogs.nejm.org](https://blogs.nejm.org/cardioexchange/2011/08/25/numbers-traps-in-clinical-practice/) (11/60 correct; 27/60 said 95%). The 2% figure was recomputed: 1/(1+0.05×999) = 1.96%. Details confirmed through Manrai AK et al., "Medicine's uncomfortable relationship with math", *JAMA Intern Med* 2014;174:991–993, [pmc.ncbi.nlm.nih.gov](https://pmc.ncbi.nlm.nih.gov/articles/PMC4955674) (20 students, 20 house officers and 20 attending physicians at Harvard Medical School). The original 1978 paper itself was not seen; these figures rest on secondary accounts.. (Two or more sources agree (secondary))
- Puzzle arithmetic: 9 ÷ 108 ≈ 8%; Try this 90/(90+90) = 50%. Source: Calculated for the book. (Calculated)

Health check:

- Disease and test are labelled invented; numbers illustrative. Source: Text (labelled on puzzle page). (Confirmed)
- 1% prevalence, 90% sensitivity, 10% false-positive rate gives PPV of 9/108, about 8%. Source: Computed; matches standard Bayes/PPV method (e.g. NHS screening leaflets present results per 1,000). (Confirmed)
- "a positive screening result is usually followed by a confirmatory test". Source: UK National Screening Committee, "Screening tests: false positives and negatives", [gov.uk](https://www.gov.uk/government/publications/nhs-population-screening-explained) (this link no longer resolved when checked on 17 October 2026) (screening identifies people needing further tests). (Confirmed)
- Casscells et al. 1978: 60 participants, commonest answer 95%, correct about 2%, 11 of 60. Source: Casscells, Schoenberger, Graboys, NEJM 1978;299:999; replication Manrai et al. 2014, [pmc.ncbi.nlm.nih.gov](https://pmc.ncbi.nlm.nih.gov/articles/PMC4955674). (Confirmed (secondary; original paper not seen))

#### 2. The praise that backfired

- Galton described the effect in 1885 (BA Section H address, *Nature* 24 Sept 1885) and set out the evidence in 1886: "Regression towards mediocrity in hereditary stature", *J. Anthrop. Inst.* 15 (1886) 246–263; 928 adult children (Table I) of 205 mid-parents. Source: Galton's scan: [pdf.galton.org](https://pdf.galton.org/essays/1880-1889/galton-1886-jaigi-regression-stature.pdf) ; Miller, "Regression": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Miller/mathword/r/). (Two or more sources agree, corrected (draft said "in 1886" only))
- "mediocrity" meant only the middle; statistics kept "regression". Source: Miller (above). (Checked)
- Kahneman: Israeli air force flight instructors; praise vs shouting; he recognised regression. Source: Kahneman, Nobel autobiography (2002): [nobelprize.org](https://www.nobelprize.org/prizes/economic-sciences/2002/kahneman/biographical/) ; *Thinking, Fast and Slow* (2011) ch. 17. (Two or more sources agree)
- A best-tenth lesson is followed by a worse one 19 times in 20 (independent luck). Source: = 0.95 (holds for any continuous i.i.d. luck). (Calculated)
- Football: improvement after sacking a manager is easy to misattribute. Source: reasoning; no figure claimed. (General knowledge (hedged))

#### 3. The coin that owes you

- Jacob Bernoulli, *Ars Conjectandi*, Basel 1713, posthumous (died 16 Aug 1705, "eight years after his death"); theorem in Part IV; pondered "for twenty years". Source: MacTutor: [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Biographies/Bernoulli_Jacob/) ; Edwards, *Significance* 2013, quoting Sylla's translation: [jhanley.biostat.mcgill.ca](https://jhanley.biostat.mcgill.ca/bios601/Surveys/EdwardsOnBernoulli.pdf). (Two or more sources agree)
- Poisson used "loi des grands nombres" in 1835 and popularised it in his 1837 book. Source: Miller, "Law of large numbers": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Miller/mathword/l/). (Checked, corrected (draft said 1837))
- 510 heads expected, 51%; lead of 20; Ann ≈ 38%, Ben ≈ 1 in 7 billion (1.36 × 10⁻¹⁰). Source: Calculated for the book. (Calculated)
- Simulated run in the answer figure. Source: seed 5 in ; captioned "one simulated run". (Illustrative)

#### 4. Who shares a birthday?

- Usually credited to Richard von Mises, 1939 ("Über Aufteilungs- und Besetzungswahrscheinlichkeiten", *Rev. Fac. Sci. Univ. Istanbul* N.S. 4, 145–163). Source: [en.wikipedia.org](https://en.wikipedia.org/wiki/Birthday_problem) (pointer); Springer 10.1007/s11009-023-10028-z. (Two or more sources agree)
- Davenport discussed it c. 1927, unpublished, because he could not believe it was new ("is said to"). Source: Coxeter, *Mathematical Recreations and Essays* 11th edn (1940) p. 45, via Good (1950) and Singmaster, *Sources* §8.B. (Checked (secondary chain))
- 0.493 / 0.507 at 23; 71% at 30; 99% at 57; 253 pairs; 253 for "your birthday". Source: Calculated for the book. (Calculated)
- Uneven real birthday distribution makes a match more likely. Source: Munford, *Am. Stat.* 31 (1977) 119 (standard reference; not fetched). (Checked (standard result: any non-uniform distribution raises the collision probability))

#### 5. Ice cream and the sea

- Galton, "Co-relations and their measurement…", *Proc. R. Soc.* 45 (1888) 135–145, read 20 Dec 1888. Source: [galton.org](https://galton.org/essays/1880-1889/galton-1888-co-relations-royal-soc/galton_corr.html) ; Miller "Correlation". (Two or more sources agree)
- Pearson's standard formula, *Phil. Trans. A* 187 (1896). Source: Miller "Correlation": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Miller/mathword/c/). (Checked)
- Pearson coined "spurious correlation", *Proc. R. Soc.* 60 (1897). Source: Miller "S": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Miller/mathword/s/). (Checked)
- Yule 1926, "Why do we sometimes get nonsense-correlations between time-series?", *JRSS* 89. Source: Miller "C", "S"; [doi:10.2307/2341482](https://doi.org/10.2307/2341482). (Two or more sources agree)
- Ice cream / rescues; fire engines; feet and reading. Source: Calculated for the book. (Illustrative / general knowledge)

#### 6. Twice as dangerous

- October 1995: CSM warned that pills containing gestodene or desogestrel carried about twice the venous-clot risk of older pills ("Dear Doctor" letter 18 Oct 1995). Source: Furedi, *Hum. Reprod. Update* 5 (1999) 621–626, [doi:10.1093/humupd/5.6.621](https://doi.org/10.1093/humupd/5.6.621) ; BMJ: [pmc.ncbi.nlm.nih.gov](https://pmc.ncbi.nlm.nih.gov/articles/PMC27461). (Two or more sources agree)
- "1 in 7,000 to 2 in 7,000" attributed to Gigerenzer's summary, not to the CSM. Source: Gigerenzer et al., *Psychol. Sci. Public Interest* 8 (2007); APS release: [psychologicalscience.org](https://www.psychologicalscience.org/news/releases/just-a-numbers-game-making-sense-of-health-statistics.html). (Checked (as Gigerenzer's framing))
- Abortions, England and Wales residents: 154,315 (1995) → 167,916 (1996). Source: ONS series via [johnstonsarchive.net](https://johnstonsarchive.net/policy/abortion/uk/ab-ukenglandwales.html) ; BMJ 1999 "rose by 9%": [pmc.ncbi.nlm.nih.gov](https://pmc.ncbi.nlm.nih.gov/articles/PMC1115440) ; Del Bono et al. (2011) [repository.essex.ac.uk](https://repository.essex.ac.uk/2573/). (Two or more sources agree)
- "Number needed to treat" proposed 1988 (Laupacis, Sackett & Roberts, *NEJM* 318:1728–33). Source: [doi:10.1056/NEJM198806303182605](https://doi.org/10.1056/NEJM198806303182605). (Checked)
- Puzzle and Try-this numbers (1→2 in 7,000; 4→3 in 1,000; 1,000 screened). Source: Calculated for the book. (Illustrative, computed)

Health check:

- Puzzle baseline "1 woman in 7,000" a year on the older pill (labelled illustrative). Source: Current MHRA (2014) figures: about 2 per 10,000 non-users, 5–7 per 10,000 on levonorgestrel/norethisterone/norgestimate pills, 9–12 per 10,000 on desogestrel/gestodene/drospirenone, i.e. about 1 in 1,700 vs 1 in 1,000. GPnotebook table citing MHRA, [gpnotebook.com](https://gpnotebook.com/pages/gynaecology/absolute-risk-of-venous-thromboembolism-dvt-/-pe-according-to-coc) ; Pulse on MHRA advice, [pulsetoday.co.uk](https://www.pulsetoday.co.uk/?p=22547) . The puzzle's real-world scale is roughly 4x too low; a reader could take 1 in 7,000 as today's risk. The relative risk (1.5–2x) is right.. (Text revised after this check)
- CSM, Oct 1995: gestodene/desogestrel pills about twice the venous-clot risk. Source: Furedi, Hum Reprod Update 1999, [doi:10.1093/humupd/5.6.621](https://doi.org/10.1093/humupd/5.6.621). (Confirmed)
- Gigerenzer's "1 in 7,000 to 2 in 7,000" summary. Source: Gigerenzer et al., Psychol Sci Public Interest 2007. (Confirmed (correctly attributed as his framing))
- "Many women stopped taking the pill"; abortions 154,315 (1995) to 167,916 (1996), linked "in part" to the scare. Source: BMJ 1999, [pmc.ncbi.nlm.nih.gov](https://pmc.ncbi.nlm.nih.gov/articles/PMC1115440) ; ONS series (main source table). (Confirmed)
- Absolute risk small compared with what clinicians emphasise: clot risk in pregnancy exceeds pill risk. Source: Vinogradova et al., BMJ 2015, [pmc.ncbi.nlm.nih.gov](https://pmc.ncbi.nlm.nih.gov/articles/PMC4444976) ; Pharmaceutical Journal summary, [pharmaceutical-journal.com](https://pharmaceutical-journal.com/article/news/newer-contraceptive-pills-associated-with-higher-risk-of-clots). (Text revised after this check)
- NNT proposed 1988. Source: Laupacis et al., NEJM 1988;318:1728. (Confirmed)
- Try this: screening 4 to 3 per 1,000 over ten years, 1,000 screened per death prevented. Source: Computed. (Confirmed (illustrative; mirrors Gigerenzer's breast-screening example))

#### 7. Where to put the armour

- Wald, Statistical Research Group, Columbia; eight memoranda (1943) on estimating vulnerability from damage on survivors. Source: CNA reprint CRC 432 (1980): [apps.dtic.mil](https://apps.dtic.mil/sti/pdfs/ADA091073.pdf) ; Casselman, "The Legend of Abraham Wald", AMS Feature Column (June 2016): [ams.org](https://www.ams.org/samplings/feature-column/fc-2016-06). (Two or more sources agree)
- Wallis recalled the military's inclination to protect the most-hit parts. Source: Wallis, "The Statistical Research Group, 1942–1945", *JASA* 75 (1980) 320–330 and rejoinder. (Checked)
- Popular "armour the engines" telling goes beyond the record; the dotted-bomber picture was drawn in 2016 †. Source: Casselman (above); [commons.wikimedia.org](https://commons.wikimedia.org/wiki/File:Survivorship-bias.png) (McGeddon, 12 Nov 2016). († labelled)
- Old buildings / successful firms. Source: reasoning. (General knowledge)

Health check:

- Try this: survey of 80-year-old smokers is biased because damaged smokers died earlier. Source: Doll et al., BMJ 2004 (50-year British doctors study: smokers die ~10 years earlier), [bmj.com](https://www.bmj.com/content/328/7455/1519). (Confirmed)

#### 8. Better in every group

- Pearson and colleagues 1899; Yule 1903 (*Biometrika* 2); Simpson 1951 (*JRSS B* 13, 238–241); named by Blyth 1972 (*JASA* 67). Source: Miller "S": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Miller/mathword/s/). (Checked, corrected (draft credited Yule alone as first))
- Berkeley, autumn 1973: 8,442 men, 44% admitted; 4,321 women, 35%; little department-level bias against women; women applied more to departments with low admission rates. Source: Bickel, Hammel & O'Connell, *Science* 187 (1975) 398–404: [homepage.stat.uiowa.edu](https://homepage.stat.uiowa.edu/~mbognar/1030/Bickel-Berkeley.pdf) (this link no longer resolved when checked on 17 October 2026). (Two or more sources agree)
- Hospital figures 635/1,000 vs 860/1,000. Source: Calculated for the book. (Illustrative, computed)

Health check:

- Hospitals and figures labelled invented. Source: Text. (Confirmed)
- Surgeon taking hardest cases can have worse crude survival while better within case-mix groups. Source: Standard case-mix adjustment rationale (e.g. NHS Digital SHMI methodology, [digital.nhs.uk](https://digital.nhs.uk/data-and-information/publications/ci-hub/summary-hospital-level-mortality-indicator-shmi)). (Confirmed)

#### 9. The shape of the curve

- de Moivre 1733 (pamphlet dated 12 Nov 1733). Source: MacTutor (gives 13 Nov): [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Biographies/De_Moivre/) ; [mathpages.com](https://mathpages.com/home/kmath642/kmath642.htm) (text gives year only). (Two or more sources agree)
- Gauss 1809, errors of astronomical measurement (*Theoria motus*). Source: standard; Pearson 1920. (Checked)
- Quetelet, chests of Scottish soldiers (1846). Source: HistData ChestSizes, citing Stigler (1986) p. 208. (Two or more sources agree)
- Pearson popularised "normal curve" and later regretted it ("…leading people to believe that all other distributions of frequency are in one sense or another 'abnormal'", *Biometrika* 13 (1920) p. 25). Source: [archive.org](https://archive.org/details/sim_biometrika_1920-10_13_1) ; Miller "N". (Two or more sources agree)
- 68 / 95 / 99.7%; 2.3% above +2 SD; heights 162 ± 7 cm illustrative. Source: Calculated for the book. (Calculated / illustrative)
- Heights within one sex close to normal; incomes right-skewed; town sizes long-tailed. Source: figure labelled schematic in alt text. (General knowledge)

Health check:

- Adult height is the sum of many genetic and environmental (childhood diet, health) influences; near-normal within one sex/generation. Source: NCD-RisC, eLife 2016 (height determinants); standard human genetics (polygenic trait). (Confirmed)
- Try this: women average 162 cm, SD 7 cm; ~2.3% above 176 cm. Source: Health Survey for England anthropometric data; computed. (Confirmed (hypothetical "if"; UK women ~162 cm, SD ~6–7 cm))

#### 10. A millionaire walks into a pub

- Galton first used "median" in English in 1881; Cournot "valeur médiane" 1843. Source: Miller "M": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Miller/mathword/m/). (Checked, corrected (draft "early 1880s"))
- "Average" from French *avarie* / Italian *avaria*, damage to ship or cargo; loss shared ("general average"). Source: [etymonline.com](https://www.etymonline.com/word/average) ; Merriam-Webster. (Two or more sources agree (Arabic root uncertain, so not mentioned))
- UK April 2025 (provisional): full-time median £39,039, mean £48,512. Source: ONS ASHE 2025 bulletin, Table 1.7a: [ons.gov.uk](https://www.ons.gov.uk/employmentandlabourmarket/peopleinwork/earningsandworkinghours/bulletins/annualsurveyofhoursandearnings/2025) — these are the provisional April 2025 figures; the 2026 survey was due in late October 2026, after the book went to press. (Checked)
- Pub £1,027,000 / £30,000; houses £612,000 / £220,000. Source: Calculated for the book. (Calculated)

#### 11. The bigger poll

- *Literary Digest* 1936: about 10 million ballots, about 2.4 million returned (2,376,523); predicted Landon. Source: Lusinchi, *Social Science History* 36 (2012): [pdixon.stat.iastate.edu](https://pdixon.stat.iastate.edu/stat587/resources/president-landon-and-the-1936-literary-digest-poll-were-automobile-and-telephone-owners-to-blame.pdf) ; arXiv 2107.14166. (Two or more sources agree)
- Roosevelt won 46 of 48 states. Source: Lusinchi; Emory: [math.oxford.emory.edu](https://math.oxford.emory.edu/site/math117/historicalBlunders/). (Two or more sources agree)
- Gallup's far smaller sample predicted Roosevelt (sample size not stated). Source: Lusinchi (Gallup 55.7% two-party). (Checked)
- Mailing lists slanted, nonresponse the bigger problem. Source: Squire, *POQ* 52 (1988) 125–133; Lusinchi 2012. (Two or more sources agree)
- ±3.1 at 1,000; ±2.2 at 2,000; ÷√2. Source: Calculated for the book. (Calculated)

#### 12. What p = 0.03 means

- Pearson 1900 chi-squared P values. Source: *Phil. Mag.* (5) 50: 157–175. (Checked)
- Fisher 1925, "convenient" to take P = .05 as a limit. Source: *SMRW* ch. III §12: [psychclassics.yorku.ca](https://psychclassics.yorku.ca/Fisher/Methods/chap3.htm). (Checked (verbatim))
- ASA 2016 statement: p-values do not measure the probability that the hypothesis is true. Source: Wasserstein & Lazar, *Am. Stat.* 70 (2016) 129–133, [doi:10.1080/00031305.2016.1154108](https://doi.org/10.1080/00031305.2016.1154108). (Checked)
- 20 × 0.05 = 1; 1 − 0.95²⁰ = 64%. Source: Calculated for the book. (Calculated)

Health check:

- Trial labelled invented; definition C (two-sided p-value under no effect). Source: ASA Statement on p-values 2016, [doi.org](https://doi.org/10.1080/00031305.2016.1154108). (Confirmed)
- A large trial can give a tiny p for a trivial benefit; a small trial can miss a real effect. Source: ASA 2016 (principles 5, 6). (Confirmed)
- 20 independent null tests at 0.05: ~1 expected, 64% chance of at least one. Source: Computed (1 − 0.95^20 = 0.642). (Confirmed)

#### 13. The cold cure

- Lind, 20 May 1747, HMS *Salisbury*, twelve scurvy patients, six pairs, oranges and lemons fastest; *Treatise* 1753. Source: James Lind Library: [jameslindlibrary.org](https://www.jameslindlibrary.org/lind-j-1753/). (Two or more sources agree)
- Lind did not draw the clear conclusion now drawn. Source: JLL (above). (Checked (standard historiography; JLL commentary))
- Colds clear up "within a week or two". Source: NHS: [nhs.uk](https://www.nhs.uk/conditions/common-cold/) ("feel better in about 1 to 2 weeks"). (Checked, corrected (draft "about a week"))

Health check:

- "Most colds clear up within a week or two whatever you do". Source: NHS "Common cold" (feel better in about 1 to 2 weeks), [nhs.uk](https://www.nhs.uk/conditions/common-cold/) ; nidirect (7–10 days adults, cough may last 2–3 weeks), [nidirect.gov.uk](https://www.nidirect.gov.uk/conditions/common-cold). (Confirmed)
- Remedy X invented; control group comparison. Source: Text; standard trial design. (Confirmed)
- Lind, 20 May 1747, HMS Salisbury, 12 sailors, 6 pairs, oranges and lemons fastest. Source: James Lind Library, [jameslindlibrary.org](https://www.jameslindlibrary.org/lind-j-1753/). (Confirmed)
- Try this: placebo controls for expectation and allows blinding of assessors. Source: Cochrane Handbook ch. 8 (performance/detection bias), [training.cochrane.org](https://training.cochrane.org/handbook). (Confirmed)

#### 14. The impressive trial

- Fisher argued for randomisation at Rothamsted in the 1920s (1926 "The arrangement of field experiments"); *The Design of Experiments* 1935. Source: standard bibliography; JLL. (Checked)
- MRC streptomycin trial, *BMJ* 30 Oct 1948, Bradford Hill, random numbers in sealed envelopes; deaths by six months 4/55 vs 14/52. Source: JLL: [jameslindlibrary.org](https://www.jameslindlibrary.org/medical-research-council-1948b/) ; Senn, *Dicing with Death*. (Two or more sources agree)
- Called "a landmark", not "the first RCT". Source: JLL on Amberson 1931 and Bell 1941. (Wording follows the fact-check)

Health check:

- Doctor-chosen allocation causes selection bias; random allocation prevents systematic differences "measured or not". Source: Cochrane Handbook ch. 8 (selection bias, allocation concealment). (Confirmed)
- "2,000 patients are enough for many trials". Source: Standard power considerations. (Confirmed)
- MRC 1948 streptomycin trial: sealed envelopes; 4/55 vs 14/52 deaths at six months. Source: MRC, BMJ 1948;2:769; James Lind Library, [jameslindlibrary.org](https://www.jameslindlibrary.org/medical-research-council-1948b/). (Confirmed)
- Try this: predictable allocation lets recruiters steer; concealed random draw prevents it. Source: Schulz et al., JAMA 1995 (inadequate concealment exaggerates effects). (Confirmed)

#### 15. Three doors

- Selvin, *Am. Stat.* 29(1) Feb 1975 p. 67; named in 29(3) Aug 1975 p. 134. Source: [en.wikipedia.org](https://en.wikipedia.org/wiki/Monty_Hall_problem) ; [tomrocksmaths.com](https://tomrocksmaths.com/2021/02/27/monty-hall-origins). (Two or more sources agree)
- Monty Hall hosted *Let's Make a Deal*. Source: [en.wikipedia.org](https://en.wikipedia.org/wiki/Let%27s_Make_a_Deal). (Two or more sources agree)
- vos Savant, *Parade* 9 Sept 1990; ~10,000 letters, most disagreeing; close to 1,000 signed with PhDs. Source: Tierney, *NYT* 21 July 1991: [academicweb.nd.edu](https://academicweb.nd.edu/~rwilliam/stats1/appendices/Monty-NYTimes.pdf). (Checked)
- 1/3, 2/3; 99/100. Source: Calculated for the book. (Calculated)

#### 16. A prize five times the stake

- Pascal–Fermat letters, 1654, problem of points; Pascal's 2–1 case. Source: [york.ac.uk](https://www.york.ac.uk/depts/maths/histstat/pascal.htm). (Checked)
- Huygens 1657, first printed probability book; his "value" (Dutch *waerde*); Latin translation by van Schooten called it *expectatio*. Source: Miller "E": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Miller/mathword/e/) ; arXiv 2407.13312. (Two or more sources agree, corrected (draft implied Huygens's own term))
- UK National Lottery: 55p in prizes per £1, 1 Apr 2024–31 Mar 2025. Source: Gambling Commission: [gamblingcommission.gov.uk](https://www.gamblingcommission.gov.uk/statistics-and-research/publication/pence-per-pound-breakdown-of-national-lottery-sales-1-april-2024-to-31-march). (Checked (official))
- 83p, 17p; £15/£5. Source: Calculated for the book. (Calculated)

#### 17. The weed on the pond

- Malthus, *Essay on the Principle of Population*, 1798, published anonymously; "geometrical ratio" vs "arithmetical ratio". Source: Liberty Fund OLL, 1798 first edition: [oll.libertyfund.org](https://oll.libertyfund.org/titles/malthus-an-essay-on-the-principle-of-population-1798-1st-ed) ; Garrison–Morton (historyofmedicine.com) on anonymous first edition, name added in 1803. (Two or more sources agree)
- "Exponent" introduced by Michael Stifel in 1544 (*Arithmetica integra*). Source: Jeff Miller, *Earliest Known Uses of Some of the Words of Mathematics*, "E" (MacTutor mirror): [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Miller/mathword/e/) , quoting Stifel and citing Smith, *History of Mathematics* vol. 2, p. 521. (Checked)
- Latin *exponere*, "to put forth". Source: Lewis & Short s.v. *expono*; etymonline page returned 403 and was not read. (Checked (standard dictionary sense: ex- "out" + ponere "to put"))
- Chessboard legend: told in many versions, no identifiable inventor †. Source: Wikipedia, "Wheat and chessboard problem" (earliest firm record Ibn Khallikān, 13th c.) and "Sissa (mythical brahmin)" ("mainly regarded as a myth"). The text names no source date for the legend.. (Two or more sources agree)
- 1.07^10 ≈ 1.97, ^20 ≈ 3.87, ^30 ≈ 7.61; exact doubling time 10.24 years; 2^63 vs 2^63 − 1. Source: Calculated for the book. (Calculated)

#### 18. The first digit

- Newcomb 1881, worn early pages of logarithm tables (*Am. J. Math.* 4, 39–40). Source: [ams.org](https://www.ams.org/featurecolumn/archive/newcomb.html). (Two or more sources agree)
- Benford, GE physicist, 1938, 20,229 numbers from 20 sources; 30.6% begin with 1 on the "Average" row. Source: Benford, *Proc. Am. Phil. Soc.* 78 (1938) 551–572; table: [archive.lib.msu.edu](https://archive.lib.msu.edu/crcmath/math/math/b/b099.htm). (Two or more sources agree)
- Theory log₁₀(2) = 30.1%; 9,000→10,000 = 11%. Source: Calculated for the book. (Calculated)
- Audit use; mismatch is not proof. Source: [en.wikipedia.org](https://en.wikipedia.org/wiki/Benford%27s_law) (pointer). (Checked (Nigrini) / courts claim dropped)

#### 19. One in a million

- Diaconis & Mosteller 1989, "law of truly large numbers"; quotation verbatim. Source: *JASA* 84 (1989) 853–861: [stat.berkeley.edu](https://www.stat.berkeley.edu/~aldous/157/Papers/diaconis_mosteller.pdf). (Checked)
- Evelyn Marie Adams, New Jersey lottery 1985 and 1986; "1 in 17 trillion"; Samuels & McCabe "better than 1 in 30" within four months. Source: same paper. (Checked)
- 60 a day, ~22,000 a year; 81% for five heads in a row in 100 tosses. Source: Calculated for the book. (Calculated)

#### 20. A one-in-a-million match

- Thompson & Schumann named it, 1987 (*Law Hum. Behav.* 11, 167–187). Source: [faculty.sites.uci.edu](https://faculty.sites.uci.edu/wthompson/publications). (Two or more sources agree)
- Sally Clark convicted 1999 of murdering her two baby sons (died 1996 and 1998). Source: *R v Clark* [2003] EWCA Crim 1020: [caselaw.nationalarchives.gov.uk](https://caselaw.nationalarchives.gov.uk/ewca/crim/2003/1020). (Checked)
- "1 in 73 million" by squaring 1 in 8,543. Source: judgment; 8,543² = 72,982,849. (Two or more sources agree)
- RSS objections: independence assumption; double murder may be rarer still. Source: RSS statement 23 Oct 2001 (phrases via secondary); RSS President's letter to the Lord Chancellor, 23 Jan 2002: [rss.org.uk](https://www.rss.org.uk/RSS/media/File-library/Membership/Sections/2020/Sally-Clark-RSS-letter-2002.pdf). (Checked)
- Quashed 2003 chiefly because microbiology evidence was not disclosed. Source: judgment (above). (Checked)
- Died 2007 (day differs between sources, so only the year is given). Source: [en.wikipedia.org](https://en.wikipedia.org/wiki/Sally_Clark). (Two or more sources agree)
- ~10 other matches among 10 million; 1 in 11. Source: Calculated for the book. (Calculated, illustrative)

#### 21. Six throws, one six

- Pascal to Fermat, 29 July 1654: de Méré's two dice bets; complaint that arithmetic contradicts itself. Source: [york.ac.uk](https://www.york.ac.uk/depts/maths/histstat/pascal.htm). (Checked)
- Antoine Gombaud; "Chevalier de Méré" a pen-name, not a title. Source: [en.wikipedia.org](https://en.wikipedia.org/wiki/Antoine_Gombaud) ; [universalis.fr](https://www.universalis.fr/encyclopedie/antoine-mere/). (Checked, corrected)
- 66.5%; 51.8%; 49.1%; 25 throws to pass half; 26% for 1-in-100 a year over 30 years. Source: Calculated for the book. (Calculated)

### B. Number, shape and logic (puzzles 22–43)

#### 22. Divide by nothing

- Babylonian scribes, from about the 4th century BC, marked an empty place *inside* a number with a sign, but did not treat it as a number. Source: MacTutor, "A history of Zero": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/). (Checked (corrected: draft draft said "Seleucid"; MacTutor dates it c. 400 BC; form varied))
- Brahmagupta, 628 (*Brāhmasphuṭasiddhānta*): number minus itself is zero; times zero is zero; 0 ÷ 0 = 0 (rejected later). Source: MacTutor, Brahmagupta: [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Biographies/Brahmagupta/) ; HistTopics/Zero (above). (Two or more sources agree)
- *śūnya* → Arabic *ṣifr* (a translation) → Fibonacci *zephirum* (1202) → Italian → *zero*; *cipher* from the same Arabic word. Source: Miller, "Z": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Miller/mathword/z/) ; HistTopics/Zero. (Two or more sources agree)
- 7 ÷ 0 undefined; 0 ÷ 0 indeterminate; #DIV/0!; zero is even. Source: reasoning. (General knowledge)
- 205 vs 25. Source: Calculated for the book. (Calculated)

#### 23. Below zero, twice

- *Nine Chapters* ch. 8 (compiled by about the 1st century AD) gives rules for adding and subtracting positive and negative numbers. Source: MacTutor, Nine Chapters: [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/HistTopics/Nine_chapters/) ; Shen Kangshen et al. translation (1999) quoted in bibnum.publimath.fr/ACF/ACF11039.pdf. (Two or more sources agree)
- Red and black rods are in Liu Hui's commentary of 263, not the main text. Source: as above. (Checked, corrected (many popular sources put them in the main text))
- Brahmagupta 628: fortunes and debts; product of two debts is a fortune. Source: MacTutor, Brahmagupta. (Checked)
- Stifel, *numeri absurdi*, *Arithmetica integra* (1544). Source: Miller, "N": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Miller/mathword/n/). (Two or more sources agree)
- Maseres, *Dissertation on the Use of the Negative Sign in Algebra* (1758). Source: MacTutor, Maseres: [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Biographies/Maseres/). (Checked)
- (−3)×(−5) = 15 from the distributive law; −3 °C → 5 °C is 8 degrees; Try: 11. Source: Calculated for the book. (Calculated)

#### 24. Multiply and add one

- Euclid, *Elements* IX.20, c. 300 BC: "Prime numbers are more than any assigned multitude of prime numbers" (Heath). Source: Joyce's Euclid: [mathcs.clarku.edu](https://mathcs.clarku.edu/~djoyce/elements/bookIX/propIX20.html). (Two or more sources agree)
- Euclid's proof uses three primes and is mostly direct, not by contradiction. Source: Hardy & Woodgold, *Math. Intelligencer* 31(4) (2009) (citation confirmed, not read in full). (Checked (nuance: one sub-step is by contradiction; text says "mostly directly"))
- Unique factorisation first clearly stated and proved in its modern form by Gauss, 1801. Source: Gauss, *Disquisitiones* art. 16; Agargün & Özkan, *Hist. Math.* 28 (2001). (Checked (wording hedged: al-Fārisī c. 1300 has a claim to the existence part))
- 30,031 = 59 × 509; remainders 1; earlier products 3, 7, 31, 211, 2,311 prime; Try 16 = 2⁴. Source: Calculated for the book. (Calculated)

#### 25. More fractions?

- Galileo, *Two New Sciences* (1638): squares pair with whole numbers; "greater"/"less" do not apply to infinite quantities. Source: Galileo, First Day (via [en.wikipedia.org](https://en.wikipedia.org/wiki/Galileo%27s_paradox) as pointer). (Checked)
- Cantor: letters to Dedekind 1873 (fractions listable; 7 Dec 1873 reals not); 1874 paper (Crelle 77). Source: [en.wikipedia.org](https://en.wikipedia.org/wiki/Georg_Cantor%27s_first_set_theory_article) ; MacTutor Cantor. (Two or more sources agree)
- Diagonal argument 1891 (*Jahresbericht der DMV* 1). Source: as above. (Checked)

#### 26. A square corner

- Plimpton 322: clay tablet from Iraq (probably Larsa), about 1800 BC; 15 rows; short side and diagonal of right triangles with whole-number sides, e.g. 119 and 169 (third side 120). Source: MacTutor, "Pythagoras's theorem in Babylonian mathematics": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_Pythagoras/) ; Columbia Magazine, "Babylon revisited": [magazine.columbia.edu](https://magazine.columbia.edu/article/babylon-revisited) (Robson c. 1822–1784 BC). (Two or more sources agree)
- Purpose debated: table of triples (Neugebauer & Sachs 1945); teacher's aid built from reciprocal pairs (Robson 2001, 2002); 2017 "exact sexagesimal trigonometry" claim (Mansfield & Wildberger) criticised by specialists. Source: Scientific American blog (E. Lamb, 2017): [blogs.scientificamerican.com](https://blogs.scientificamerican.com/roots-of-unity/dont-fall-for-babylonian-trigonometry-hype/) ; Robson, *Hist. Math.* 28 (2001) 167–206. (Checked (text says "criticised", not "refuted"))
- Pythagoras c. 570–c. 490 BC (C: draft said 495); left no writings; linked with the theorem only by much later writers. Source: Stanford Encyclopedia of Philosophy, "Pythagoras": [plato.stanford.edu](https://plato.stanford.edu/entries/pythagoras/). (Two or more sources agree)
- Oldest surviving proof: Euclid I.47. Source: standard. (Checked)
- Egyptian 3-4-5 rope †. Source: Wikipedia "Rope stretcher" (pointer); proofwiki. († dagger: 19th-century guess (Moritz Cantor), no ancient evidence)
- Backlog said "Babylonian tablet Plimpton 322 (c. 1800 BC)" as the *origin* of the theorem. Source: the text says the relation was "in use" over a thousand years before Pythagoras; it does not claim the tablet states a theorem. (Corrected reframed)
- 36+64=100; 4,5,6 → 82.8°; 5,5,7 → 88.85°; wall 10 m → 20 cm; 20,21,29. Source: Calculated for the book. (Calculated)

#### 27. Round the tin

- Archimedes, *Measurement of a Circle* Prop. 3, 96-sided polygons: 3 10/71 < π < 3 1/7. Source: MacTutor, "Pi through the ages": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/HistTopics/Pi_through_the_ages/). (Checked)
- Lambert's irrationality proof published 1768. Source: MacTutor Lambert; Pi through the ages. (Checked (corrected: "1761" is the year of presentation in some accounts; text says "published in 1768"))
- Zu Chongzhi (429–501): 3.1415926 < π < 3.1415927; 355/113. Source: MacTutor, Zu Chongzhi. (Checked)
- Jones 1706; Euler made π standard. Source: Miller, symbols for constants: [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/constants/). (Two or more sources agree)
- 22/7 − π = 0.00126 (1 part in 2,484); tin gap 0.09 mm; π − 3.14 = 0.00159. Source: Calculated for the book. (Calculated)

#### 28. Up and down

- Under Augustus, a tax of one hundredth on goods sold at auction (*centesima rerum venalium*). Source: Smith's *Dictionary of Greek and Roman Antiquities*, "Centesima": [penelope.uchicago.edu](https://penelope.uchicago.edu/Thayer/E/Roman/Texts/secondary/SMIGRA*/Centesima.html). (Checked (corrected: "under", not "introduced by"; Tacitus *Ann.* 1.78))
- *per cento*; % sign from merchants' abbreviations, early form in a 15th-century Italian manuscript; later "per o/o". Source: [en.wikipedia.org](https://en.wikipedia.org/wiki/Percent_sign) (pointer to Smith). (Checked (secondary; Smith *Rara Arithmetica*, *History of Mathematics* vol. 2 not read directly; text says "15th century" rather than "c. 1425"))
- £75 either order; 20% cut needs 25%; 10% up/down → 99%. Source: Calculated for the book. (Calculated)

#### 29. Interest, more often

- Jacob Bernoulli, paper published 1690 (*Acta Eruditorum*): continuous compounding stays between 2 and 3. Source: Pulskamp translations: [gtfp.cs.rhul.ac.uk](http://gtfp.cs.rhul.ac.uk/pulskamp/JakobBernoulli/JakobB.html) ; MacTutor "The number e": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/HistTopics/e/). (Checked, corrected (draft said 1683; the 1685 *Journal des Sçavans* problems were about dice))
- Euler wrote e in a manuscript of c. 1727–28; in print 1736 (*Mechanica*); 23 decimals in 1748 (*Introductio*). Source: Miller, constants; quadrivium.info/MathInt/Notes/Euler_e.pdf. (Two or more sources agree, corrected (Britannica/MacTutor "18 decimals" is wrong; *Introductio* ch. VII shows 23))
- UK lenders must quote APR. Source: Standard reference fact or general knowledge; no single source recorded. (General knowledge (Consumer Credit Act regime))
- £2.61 monthly, £2.71 daily, 2.71828; 1%/month = 12.68%. Source: Calculated for the book. (Calculated)

#### 30. The price of a loaf

- Pacioli, *Summa de arithmetica* (Venice 1494): keep 72 in mind and divide by the rate; no explanation; earliest surviving statement. Source: [en.wikipedia.org](https://en.wikipedia.org/wiki/Rule_of_72). (Checked (secondary only: f. 181 no. 44 "tieni per regola 72"; confirmed through Vacca G., "A Napierian logarithm before Napier" (1915), reprinted at MacTutor: [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Extras/Pacioli_logarithm/) , which quotes the Italian; no scan of the 1494 folio was seen, so this is a secondary source))
- 1.08⁹ = 1.999; 9.006 years; 8p → 16p in year 10; 3% → 23.4; 6% → 11.9. Source: Calculated for the book. (Calculated)

#### 31. One step up the scale

- Napier, *Mirifici Logarithmorum Canonis Descriptio*, Edinburgh 1614; coined "logarithm", usually read as "ratio-number" (he never explained it). Source: Miller "L": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Miller/mathword/l/) ; MacTutor Napier. (Two or more sources agree (corrected: "reckoning" gloss dropped))
- Richter 1935, *BSSA*, worked out with Gutenberg. Source: Britannica; USGS. (Two or more sources agree)
- One step ≈ ×10 recorded amplitude, ≈ ×32 energy; moment magnitude 1979 (Hanks & Kanamori) used for large quakes. Source: USGS earthquake-magnitude FAQ; Hanks & Kanamori, *JGR* 84 (1979) 2348. (Two or more sources agree)
- Bel/decibel named after Bell; name adopted by Bell System engineers 1928 (announced 1929). Source: W. H. Martin, *BSTJ* Jan 1929; Wikipedia "Decibel". (Two or more sources agree)
- 10 dB = ×10 intensity; pH 4 vs 6 = ×100; lemon/coffee ×1,000. Source: Calculated for the book. (Calculated)

#### 32. The giant

- Galileo, *Two New Sciences* (Leiden 1638), Day Two: ships, palaces, trees, animals cannot be enlarged without limit; bone three times as long drawn much thicker. Source: [spirasolaris.ca](https://www.spirasolaris.ca/sbb6Second_Day.pdf). (Checked (primary text, Crew & de Salvio trans., pp. 130–131))
- Haldane, "On Being the Right Size", *Harper's*, March 1926: mouse/rat/man broken/horse splashes (paraphrase). Source: [cs.utexas.edu](https://www.cs.utexas.edu/~witchel/S25-380L/papers/right-size.html). (Two or more sources agree)
- Ant survives falls; elephant legs proportionally stout. Source: Standard reference fact or general knowledge; no single source recorded. (General knowledge)
- ×8 weight, ×4 area, ×2 stress; ×10 at scale 10; sugar 24 → 48 cm². Source: Calculated for the book. (Calculated)

#### 33. There and back

- Archytas fragment, preserved by Porphyry, defines arithmetic, geometric and "subcontrary, which they call harmonic" means. Source: SEP, "Archytas": [plato.stanford.edu](https://plato.stanford.edu/entries/archytas/) ; MacTutor Archytas. (Two or more sources agree (corrected: the "renamed from subcontrary" story is Iamblichus, later))
- Who named it is disputed among ancient writers. Source: Heath, *HGM* I p. 85. († stated as such)
- 45 + 30 min; 48 mph; cyclist 15 mph. Source: Calculated for the book. (Calculated)

#### 34. Finding one name

- al-Khwārizmī worked in Baghdad, early 9th century, "traditionally linked" with the House of Wisdom (formal post undocumented, per the AI review 2.16); book on Indian numerals; Arabic lost; Latin version opens "Dixit Algorizmi". Source: MacTutor al-Khwārizmī: [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Biographies/Al-Khwarizmi/) ; Miller "A": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Miller/mathword/a/). (Two or more sources agree)
- "algorism" → "algorithm" shaped by Greek *arithmos*; "algebra" from *al-jabr*. Source: Miller "A" (OED). (Two or more sources agree)
- Euclid's algorithm (VII.1–2) among the oldest still in use. Source: Knuth, *TAOCP* vol. 2 §4.5.2 (quoted). (Two or more sources agree)
- 2²⁰ = 1,048,576; at most 976 after 10 looks, 1 after 19 (C: draft said 977); 30 looks for a billion; gcd(1071, 462) = 21. Source: Calculated for the book. (Calculated)

#### 35. Five switches

- Leibniz, "Explication de l'arithmétique binaire", *Mémoires* for 1703 (printed 1705); worked on binary since at least 1679. Source: historyofinformation.com/detail.php?id=395; arXiv 1612.06373. (Two or more sources agree)
- Bouvet's letter (sent 1701) reached Leibniz in 1703 with the Fu Xi hexagram order; read as binary 0–63. Source: arXiv 1612.06373; Swarthmore Leibniz–Bouvet edition. (Two or more sources agree)
- No evidence the Chinese authors meant arithmetic. Source: Maitre, *Science in Context* (abstract). (Checked)
- Harriot binary in unpublished papers about a century earlier (early 1600s). Source: MacTutor, "Harriot's binary numbers": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Extras/Harriot_binary_numbers/). (Two or more sources agree)
- 2⁵ = 32; 10110 = 22; 1011 = 11. Source: Calculated for the book. (Calculated)

#### 36. Seven bridges

- Euler's paper read to the St Petersburg Academy 1735/36, printed 1741 (vol. for 1736). Source: Grötschel & Yuan, *Documenta Math.* Extra Vol. ISMP (2012) 43–50; Sachs, Stiebitz & Wilson, *J. Graph Theory* 12 (1988). (Two or more sources agree (corrected: "1736" in draft; the 26 Aug 1735 date may be a misprint))
- Bridges 5, 3, 3, 3; impossibility proof; sufficiency stated without proof; Hierholzer's proof published 1873 after his death. Source: as above. (Two or more sources agree)
- Euler used a map and letters, not dots and lines. Source: Robin Wilson (quoted). (Checked)
- Königsberg now Kaliningrad; war destroyed several bridges. Source: [kconrad.math.uconn.edu](http://kconrad.math.uconn.edu/kaliningrad/). (Checked (number of survivors conflicts; no number printed))
- Degrees 5,3,3,3; house figure odd corners = 2 bottom. Source: Calculated for the book. (Calculated)

#### 37. Pieces of a circle

- Pólya 1919 conjecture. Source: MathWorld, "Pólya Conjecture": [mathworld.wolfram.com](https://mathworld.wolfram.com/PolyaConjecture.html). (Two or more sources agree)
- Haselgrove 1958 disproof without explicit counterexample; Lehman 1960 (906,180,359); Tanaka 1980: smallest 906,150,257. Source: MathWorld; Wikipedia "Pólya conjecture". (Two or more sources agree (corrected: draft implied Haselgrove found the first failure))
- "Moser's circle problem", named after Leo Moser; now a standard warning (the text no longer says Moser himself used it so). Source: MathWorld, "Circle Division by Chords"; arXiv 1701.08155. (Two or more sources agree)
- 1, 2, 4, 8, 16, 31, 57; C(6,4)=15; even hexagon 30; n²+n+41 prime 0–39, 1,681 = 41². Source: Calculated for the book. (Calculated)

#### 38. Measuring the coast

- Richardson; published posthumously 1961 (*General Systems Yearbook* 6). Source: Mandelbrot 1967 (below); Wikipedia "Coastline paradox". (Two or more sources agree)
- Spain–Portugal border figures 987 km and 1,214 km (which country gave which is disputed; not stated). Source: Wikipedia "Coastline paradox"; futilitycloset.com/2013/12/02/the-richardson-effect/. (Two or more sources agree)
- Mandelbrot, *Science* 156 (1967) 636–638; west coast of Britain D ≈ 1.25. Source: gsp.humboldt.edu/olm_2021/Courses/GSP_510/Articles/Mandelbrot1967.pdf. (Two or more sources agree (paper read))
- "fractal" coined 1975, Latin *fractus*. Source: Science News; etymonline. (Two or more sources agree)
- Halving ruler → ×2^0.25 = 1.19 ("about a fifth"); Koch (4/3)^10 = 17.8. Source: Calculated for the book. (Calculated)

#### 39. The portrait’s proportions

- Euclid: "extreme and mean ratio" (VI Def. 3), used for the pentagon (IV) and regular solids (XIII). Source: Joyce's Euclid, VI Def. 3; MacTutor "The Golden ratio": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/HistTopics/Golden_ratio/). (Two or more sources agree)
- Pacioli, *De divina proportione* (1509), Leonardo's illustrations; for buildings he recommended (Vitruvian) simpler proportions. Source: Livio via Wikipedia "Golden ratio"; A. J. Cain (2026) blog: ajcain.codeberg.page/posts/2026-02-21-golden-ratio-myth-development.html. (Two or more sources agree, corrected (popular claim that he urged it on artists traces to Montucla 1799))
- "Golden section" usually traced to Martin Ohm's textbook (1835). Source: Fowler, *Fibonacci Quarterly* 20 (1982): mathstat.dal.ca/FQ/Scanned/20-2/fowler.pdf. (Checked (corrected: "first used" is doubtful; possible earlier use in Gehler's *Physikalisches Wörterbuch* not seen; text says "popularised"))
- Zeising 1854, universal law; source of the modern myth. Source: Fowler; Cain; Wikipedia. (Two or more sources agree)
- Rectangle-preference tests mixed (Fechner 1876 onwards). Source: Wikipedia; Markowsky (below). (Two or more sources agree)
- Parthenon / Great Pyramid claims rest on choosing what to measure. Source: Markowsky, "Misconceptions about the golden ratio", *College Math. J.* 23 (1992) 2–19. (Two or more sources agree)
- Le Corbusier's Modulor used it deliberately. Source: Wikipedia; Fondation Le Corbusier. (Two or more sources agree)
- Golden angle in plant phyllotaxis genuine. Source: Wikipedia; Swinton et al., *R. Soc. Open Sci.* 3 (2016) 160091 (about 1 spiral count in 5 non-Fibonacci; not printed). (Checked)
- Illustrative model: 12 log-uniform lengths in 1–10 cm, 66 pairs, P(some ratio within 1% of φ) = 0.59 (100,000 trials). Source: Calculated for the book. (Calculated, labelled illustrative)
- 13/8 = 1.625, 34/21 = 1.619. Source: Calculated for the book. (Calculated)

#### 40. Two shops

- 1950: Flood and Dresher (RAND) had Alchian and Williams play 100 times. Source: SEP, "Prisoner's Dilemma": [plato.stanford.edu](https://plato.stanford.edu/entries/prisoner-dilemma/) ; Poundstone (via Wikipedia). (Two or more sources agree)
- Tucker added the prisoners' story for Stanford psychologists, 1950; name stuck. Source: SEP. (Checked (month U, not printed))
- Axelrod 1980 tournaments; Rapoport's tit for tat won. Source: Axelrod, *J. Conflict Resolution* 24 (1980). (Two or more sources agree)
- Payoffs 10/10, 3/12, 12/3, 6/6. Source: Calculated for the book. (Illustrative)

#### 41. The nosy courier

- Diffie & Hellman, Nov 1976 (*IEEE Trans. IT* 22(6)); Merkle's related work. Source: Crossref 10.1109/TIT.1976.1055638; ACM Turing Award 2015 release. (Two or more sources agree)
- RSA 1977. Source: Crossref 10.1145/359340.359342. (Two or more sources agree)
- GCHQ: Ellis 1970, Cocks 1973, Williamson 1974; made public 1997. Source: Schneier, essay 377; University of Bristol (Cocks). (Two or more sources agree (corrected: draft had "Cox"; correct is Clifford **Cocks**))
- 3,233 = 53 × 61; 2048-bit RSA modulus = 617 digits ("more than 600"). Source: Calculated for the book. (Calculated)

#### 42. Nearly the same start

- Lorenz, MIT, 1961, twelve-variable model; typed 0.506 for 0.506127. Source: APS News (Jan 2003): [aps.org](https://www.aps.org/apsnews/2003/01/lorenz-butterfly-effect-1961). (Two or more sources agree)
- 1963 simpler model ("Deterministic Nonperiodic Flow", *J. Atmos. Sci.* 20). Source: Crossref. (Checked)
- 1972 AAAS talk; butterfly title supplied by the convener (Philip Merilees). Source: Wikipedia "Butterfly effect"; arXiv 0910.2213. (Two or more sources agree)
- Poincaré 1908 (*Science et méthode*). Source: Britannica. (Two or more sources agree)
- For large weather systems, useful skill to roughly ten days; near-perfect data would push the limit to about two weeks. Source: adapt.psu.edu/ZHANG/papers/Zhangetal2019JAS.pdf. (Checked (Zhang et al., *JAS* 76 (2019)))
- Shift-map sequences; Lorenz-63 figure (σ=10, ρ=28, β=8/3; starts 1.0 and 1.0001). Source: Calculated for the book. (Calculated / illustrative)
- Error doubling every two days → ×1,024 in 20 days. Source: Calculated for the book. (Calculated, labelled illustrative)

#### 43. A lifetime of heartbeats

- Fermi at Trinity, 16 July 1945: paper scraps, blast ~40 s after, moved ~2½ m, ~10,000 tons TNT. Source: [atomicarchive.com](https://www.atomicarchive.com/resources/documents/trinity/fermi.html). (Two or more sources agree (Fermi's own report))
- Modern yield ~21–25 kt. Source: LANL 1663 (2022); Selby et al., arXiv 2103.06258. (Two or more sources agree)
- Piano tuners in Chicago "traditionally linked to" Fermi. Source: themathdoctors.org; no contemporary record. († (labelled as tradition))
- Resting heart ~70 bpm (60–80 used as range). Source: Standard reference fact or general knowledge; no single source recorded. (General knowledge)
- 2.9 billion; 2.2–3.8 billion; 947 million seconds; rubbish 100 t/person/yr = 0.27 t/day. Source: Calculated for the book. (Calculated)

Health check:

- ~70 beats a minute as a rough lifetime average; children faster, slower in sleep. Source: NHS "How do I check my pulse?" (normal resting adult 60–100 bpm), [nhs.uk](https://www.nhs.uk/common-health-questions/accidents-first-aid-and-treatments/how-do-i-check-my-pulse/) ; BHF. (Confirmed)
- ~2.9 billion beats in 80 years. Source: Computed. (Confirmed)

### C. Matter, motion and energy (puzzles 44–64)

#### 44. The cup on the table

- Aristotle: a body moves only while a mover acts on it; for projectiles, the air (*Physics* VIII.10, 266b–267a). Source: Bekker refs; Wikipedia "Inertia" (pointer). (Checked)
- Galileo: a ball on a level surface with nothing to slow it would move for ever; his "level" follows the Earth's curve (*Dialogue* 1632; *Two New Sciences* 1638, Third and Fourth Days). Source: Crew & de Salvio trans.: [archive.org](https://archive.org/details/cu31924012322701). (Two or more sources agree)
- Descartes, *Principia Philosophiae* (1644), II.37 and 39: straight-line persistence. Source: Cambridge Descartes Lexicon, "Inertia". (Checked)
- Newton, *Principia* (1687), Law I. Source: Principia, Axiomata. (Checked)
- Train 30 m/s braking 1.2 m/s², cup friction 0.05 → 28.8 and 29.5 m/s after 1 s; spanner 3.6 km. Source: Calculated for the book. (Calculated, labelled illustrative)

#### 45. Two steel balls

- Aristotle: heavier bodies fall faster (*De Caelo* I.6; *Physics* IV.8). Source: Bekker refs. (Checked)
- Philoponus (6th c.): weights differing greatly differ only slightly in time of fall. Source: Cohen & Drabkin trans.; Encyclopedia.com "John Philoponus". (Two or more sources agree)
- Stevin & de Groot, 1586, two lead balls (one ten times heavier), about 30 ft, Delft church tower (Stevin names no church). Source: Wikipedia "Delft tower experiment" (citing Sarton, *Isis* 21, 1934); DBNL Dutch text. (Checked, corrected (church))
- Galileo's inclined planes; *Two New Sciences* 1638. Source: Crew & de Salvio. (Checked)
- Pisa drop rests only on Viviani's *Racconto istorico* (written 1654, printed 1717); Galileo never mentions it †. Source: Museo Galileo: [brunelleschi.imss.fi.it](https://brunelleschi.imss.fi.it/pencil/eng/html/Space/02%20Torre.htm). († (dagger in text))
- Apollo 15, 2 Aug 1971, David Scott, hammer and falcon feather. Source: NASA NSSDC: [nssdc.gsfc.nasa.gov](https://nssdc.gsfc.nasa.gov/planetary/lunar/apollo_15_feather_drop.html). (Two or more sources agree)
- 1 kg and 10 kg steel spheres (7,850 kg/m³) are 6.2 and 13.4 cm across; from 10 m: 1.43 s in vacuum; with drag (Cd 0.47, air 1.204 kg/m³) the 1 kg ball is 1.5 cm and 1.1 ms behind. Source: Calculated for the book. (Calculated)
- Galileo's tied-stones argument (Try this). Source: *Two New Sciences*, First Day. (Checked)

#### 46. The ball that stops bouncing

- Mayer 1842, *Ann. Chem. Pharm.* 42, 233–240. Source: Wikipedia "History of energy" (pointer). (Checked)
- Joule, *Phil. Trans.* 140 (1850): 772 lb falling 1 ft warms 1 lb water by 1 °F (Joule did not say "BTU"). Source: Scan: [www2.lawrence.edu](https://www2.lawrence.edu/fast/STONEKIM/COURSES/Joule_1850.pdf). (Two or more sources agree)
- Helmholtz, *Über die Erhaltung der Kraft*, 1847. Source: as above. (Checked)
- Ball warms ≈ 0.005 °C per metre (rubber 2,000 J/kg K, illustrative); waterfall 100 m → 0.23 °C. Source: Calculated for the book. (Calculated)

#### 47. Ten molecules in a box

- Clausius 1865, *Ann. Phys. Chem.* 125, 353–400: "entropy" from τροπή, transformation, made similar to "energy". Source: Giunta, Le Moyne (Magie trans.): [web.lemoyne.edu](https://web.lemoyne.edu/giunta/Clausius.html). (Two or more sources agree)
- Boltzmann 1877 statistical interpretation. Source: standard histories. (Checked)
- S = k log W on Boltzmann's monument, Zentralfriedhof Vienna (bust 1933); form and k due to Planck. Source: Atlas Obscura; Planck's Nobel lecture. (Two or more sources agree)
- 1/1,024; 1/1,048,576; 2^100; teaspoon of milk (5 g water) ≈ 1.7×10²³ molecules; 1 in 10^(5.0×10²²). Source: Calculated for the book. (Calculated)

#### 48. Spark or splash

- Sparkler sparks "more than 1,000 °C". Source: VUMC news (2026); KSL. (Checked (conservative; sources say ≈2,000 °F ≈ 1,100 °corrected))
- Sparklers burn many children; fire services advise none for under-fives. Source: as above; UK fire and rescue services. (Checked (US injury data; UK fire-service advice))
- Black: heat vs temperature; latent heat c. 1761–62; specific heat; *Lectures* (ed. Robison) 1803. Source: Britannica "Joseph Black". (Checked (standard))
- c(iron) ≈ 450, c(water) ≈ 4,184 J/(kg K); 1 mg at 1,100 → 35 °C gives 0.48 J; 50 g tea 70 → 35 °C gives 7,322 J; ratio ≈ 15,000; iron/water heat per gram 0.11. Source: Calculated for the book. (Calculated (illustrative masses))

#### 49. How cold can it get?

- Amontons c. 1702: equal steps in temperature give equal steps in pressure; zero of "spring" ≈ −240 °C. Source: Britannica (Amontons); Wikipedia. (Two or more sources agree)
- Thomson 1848 absolute scale (zero at −∞); Joule–Thomson 1854 scale with zero near −273 °C. Source: MIT history note; Wikipedia "Kelvin". (Corrected)
- 0 K = −273.15 °C; kelvin defined via k = 1.380649×10⁻²³ J/K since 2019. Source: BIPM SI Brochure. (Checked)
- He boils 4.22 K (−268.9 °C), N₂ 77.36 K (−195.8 °C), dry ice sublimes −78.5 °C. Source: Standard reference fact or general knowledge; no single source recorded. (Standard)
- CMB 2.7255 K. Source: Fixsen 2009. (Checked)
- Atoms cooled below a billionth of a kelvin (MIT 2003 ≈ 450 pK; Bremen 2021, 38 pK effective). Source: Leanhardt et al., *Science* 301 (2003); Deppner et al., *PRL* 127 (2021). (Checked)
- Unattainability (Nernst; stated 1912). Source: arXiv 1904.11696. (Checked (not dated in text))
- 283 → 293 K is +3.5%. Source: Calculated for the book. (Calculated)

#### 50. The longest straw

- *Two New Sciences* (1638): pumps cannot lift water above 18 braccia (≈10.5 m). Source: Crew & de Salvio p. 64. (Checked)
- Torricelli's mercury tube (1643 or 1644, probably performed by Viviani; text says "in the 1640s … had a tube filled"); letter to Ricci 11 June 1644, "Noi viviamo sommersi nel fondo d'un pelago d'aria elementare". Source: Chemistry World; IMSS: [imss.fi.it](https://www.imss.fi.it/vuoto/ericci.html). (Two or more sources agree)
- Périer, Puy de Dôme, 19 Sept 1648, at Pascal's request. Source: Britannica "Pascal". (Checked (standard))
- 101,325 Pa ≈ 1.03 kg-weight per cm²; water column 10.35 m; with vapour pressure 2.339 kPa at 20 °C, 10.11 m; mercury 76.0 cm. Source: Calculated for the book. (Calculated)

#### 51. The anchor overboard

- Archimedes, *On Floating Bodies* I, Props 5 and 7. Source: Heath, *Works of Archimedes* (1897). (Checked)
- Eureka story from Vitruvius IX, about two centuries later †. Source: NYU, Rorres: [math.nyu.edu](https://www.math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html). (Two or more sources agree (dagger in text))
- 20 kg anchor: 20 L vs 2.54 L; 17.5 L; pool 10 × 5 m → 0.35 mm; iron 7.87 g/cm³. Source: Calculated for the book. (Calculated (illustrative anchor and pool))
- Ice cube melting leaves the level unchanged (fresh water). Source: Archimedes' principle. (Checked)

#### 52. A blue sky

- Tyndall, 1869: fine particles scatter faint blue light; proposed it as the cause of the sky's colour. Source: Tyndall J., "On the blue colour of the sky, the polarization of skylight, and on the polarization of light by cloudy matter generally", *Proc. R. Soc. Lond.* 17 (1869) 223–233, [doi:10.1098/rspl.1868.0033](https://doi.org/10.1098/rspl.1868.0033) (cited via Wikipedia "Rayleigh scattering"). (Checked)
- Rayleigh, 1871: scattering by particles much smaller than the wavelength varies as 1/λ⁴. Source: Strutt J.W., "On the light from the sky, its polarization and colour", *Phil. Mag.* 41 (1871) 107–120 and 274–279; Wrocław digital library summary; Physics Today (1982) on Rayleigh. (Two or more sources agree)
- Rayleigh, 1899: air molecules alone suffice. Source: Rayleigh, "On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky", *Phil. Mag.* 47 (1899) 375–384, [doi:10.1080/14786449908621276](https://doi.org/10.1080/14786449908621276). (Checked)
- Rayleigh = John William Strutt. Source: standard (Rayleigh was the 3rd Baron's title). (Checked)
- Why not violet: less violet in sunlight, lower eye sensitivity, mixture looks pale blue. Source: Smith G.S., "Human color vision and the unsaturated blue color of the daytime sky", *Am. J. Phys.* 73 (2005) 590–597. (Checked)
- (700/450)^4 = 5.86 ("nearly six times"); chart values 9.4× … 1.0×. Source: Calculated for the book. (Calculated)
- Sunset light has crossed "tens of times as much air" as at noon. Source: Kasten & Young (1989) air-mass formula gives ≈ 38 at the horizon (relative to overhead); noon Sun is rarely overhead, so "tens of times" is conservative. (Calculated)
- Moon's daytime sky is black. Source: no appreciable atmosphere; Apollo surface photography. (Checked)

#### 53. A pause on the line

- In August 1676 the Paris Observatory predicted that an emergence of Io in November would run about ten minutes late; on 9 November one did, and on 21 November Rømer explained it; ~22 min across Earth's orbit. Source: Bobis L. & Lequeux J., "Cassini, Rømer and the velocity of light", *J. Astron. Hist. Heritage* 11 (2008) 97–105, [articles.adsabs.harvard.edu](https://articles.adsabs.harvard.edu/pdf/2008JAHH...11...97B) ; Britannica "Ole Rømer"; ESA "30 September"; Observatoire de Paris. (Two or more sources agree, corrected (corrected against the full text of Bobis & Lequeux 2008))
- Io's period 1.769 d ≈ 42½ h. Source: NASA Io page. (Checked (standard))
- c = 299,792,458 m/s exactly (17th CGPM, 1983). Source: BIPM. (Checked)
- Apollo round trip ≈ 2.6 s (NASA ALSJ: 2.568 s Madrid–Moon–Madrid; 2.712 s heard in Houston). Source: [nasa.gov](https://www.nasa.gov/history/alsj/a12/A12Audio_Moon_Houston.html). (Checked)
- Moon 384,400 km → 2.56 s round trip (2.38–2.71 s perigee–apogee); Sun 499 s; Mars 3.0–22.3 min; 225 million km → 12.5 min; orbit crossing 16.6 min; half a minute is 11.7× the round trip. Source: Calculated for the book. (Calculated)

#### 54. Flash, then rumble

- Lightning channel heated to up to 50,000 °F (≈28,000 °C). Source: NWS: [weather.gov](https://www.weather.gov/safety/lightning-temperature). (Checked, corrected)
- Met Office: divide seconds by 3 for km. Source: [weather.metoffice.gov.uk](https://weather.metoffice.gov.uk/learn-about/weather/types-of-weather/thunder-and-lightning). (Checked)
- NWS: lightning can strike up to 10 miles from the storm. Source: [weather.gov](https://www.weather.gov/safety/lightning-science-thunder). (Checked)
- Mersenne and Gassendi, 1630s, gun-flash timing, ≈448 and ≈478 m/s (too high); Gassendi: cannon and musket travel equally fast. Source: Finn, *Isis* 55 (1964) 7–19; Whewell. (Two or more sources agree)
- Paris Academy 1738 ≈ 337 m/s. Source: Finn. (Checked)
- Newton too low by almost a fifth; Laplace's correction, memoir 1816. Source: Finn. (Checked, corrected (dates))
- c(20 °C) = 343.2 m/s (ideal dry air); 6 s → 2.06 km; 2.9 s/km; 4.7 s/mile; light over 2 km 6.7 µs; 15 s → 5.1 km; cricket 150 m → 0.44 s. Source: Calculated for the book. (Calculated)

#### 55. A bird on the wire

- Volta's letter to Banks, 20 March 1800, *Phil. Trans.* 90 (1800). Source: [doi:10.1098/rstl.1800.0018](https://doi.org/10.1098/rstl.1800.0018). (Checked (standard))
- Ohm, *Die galvanische Kette, mathematisch bearbeitet*, 1827. Source: MacTutor "Ohm". (Checked (standard))
- Paris 1881 congress adopted volt, ohm and ampere internationally. Source: NBS Circular 475. (Checked, corrected ("adopted"))
- Large birds electrocuted when wings bridge conductors or earthed metal; mitigation. Source: APLIC 2006; APLIC 1996. (Two or more sources agree)
- UK distribution 11 kV, rated between phases; phase to earth 11,000/√3 ≈ 6,350 V. Source: ENA. (Checked (standard), corrected after the AI review)
- UK advice: keep well clear of a fallen line (ENA/SSEN: at least 5 m; the AI review cited 10 m, so no figure is printed); call 105 in Great Britain. Source: ENA/SSEN leaflet; SSEN "Stay alive, call 105". (Two or more sources agree)
- Feet 5 cm apart, 200 A, 0.2 Ω/km → 0.002 V. Source: Calculated for the book. (Calculated, labelled illustrative)

Health check:

- Current flows only with a voltage difference; one live wire and nothing else makes no path. Source: Standard. (Confirmed (physics) — but see safety note below)
- 11 kV line, ~6,350 V phase to earth. Source: ENA; 11,000/√3. (Confirmed)
- Bird's body conducts well enough; line currents far exceed lethal levels. Source: Standard (lethal ~tens of mA). (Confirmed)
- Large birds electrocuted bridging wires or wire and earthed metal. Source: APLIC 2006. (Confirmed)
- Try this: step-potential near a fallen line; "Keep well clear and, in Great Britain, call 105". Source: ENA/SSEN: keep at least 5 m away, assume the line is live, call 105; **for danger to life or injury dial 999 and say electricity is involved**. SSEN "Stay alive, call 105", [ssen.co.uk](https://www.ssen.co.uk/news-views/2019/2019-stay-alive-call-105) (this link no longer resolved when checked on 17 October 2026) ; UK Power Networks ("if lines are down or causing significant risk to the public please call 999"), [ukpowernetworks.co.uk](https://ukpowernetworks.co.uk/help/how-do-i-get-my-power-turned-off-urgently) . Omitting 999 could delay help where someone is hurt.. (Text revised after this check)
- Physics lesson could be misread as "touching one wire is safe" for people. Source: People standing on the ground or a ladder are earthed; the text's "touches nothing else" condition covers this, and the eagle line reinforces it. No change needed if space is tight.. (Text revised after this check)

#### 56. Which pole is which?

- Gilbert, *De Magnete* (1600): terrella, Earth a great magnet; coined *electricus*. Source: Britannica; MacTutor "Gilbert". (Checked (standard))
- Gellibrand 1635: declination changes over time. Source: MacTutor "Gellibrand". (Checked (standard))
- North magnetic pole moving from Canada towards Siberia; 50–60 km/yr c. 2000–2015, now ≈ 35 km/yr. Source: BGS WMM2025 news; NOAA NCEI 2025 report. (Two or more sources agree)
- Greenwich compasses pointed to true north in September 2019, first time in ~360 years. Source: BGS; Royal Museums Greenwich. (Two or more sources agree)
- Arctic pole is physically a south magnetic pole; dip needle vertical there. Source: basic physics; BGS education pages. (Checked)

#### 57. Clocks in orbit

- Einstein 1905 (*Ann. Phys.* 17), 1907 (*Jahrbuch*), GR 1915. Source: [doi:10.1002/andp.19053221004](https://doi.org/10.1002/andp.19053221004). (Checked (standard))
- Pound–Rebka 1959 (PRL 1960), 22.6 m tower. Source: APS: [physics.aps.org](https://physics.aps.org/story/v16/st1). (Checked, corrected (height))
- Hafele–Keating Oct 1971 (*Science* 1972). Source: [doi:10.1126/science.177.4044.166](https://doi.org/10.1126/science.177.4044.166). (Checked (standard))
- NTS-2 (1977), first caesium clock in orbit, confirmed shift. Source: Ashby, *Living Rev. Relativ.* 6 (2003) 1. (Checked)
- GPS: a = 26,561.75 km (altitude 20,190 km above mean radius), v = 3.87 km/s, period 11.97 h; +45.8 (gravity) −7.2 (speed) = +38.6 µs/day; offset 4.465×10⁻¹⁰; light travels 11.6 km in 38.6 µs; 9.4 ns/day per km of height. Source: Calculated for the book. (Calculated (Ashby gives 45.7, 7.1, 38.6; differences are rounding of the ground clock's potential))

#### 58. Firing at gold

- Geiger & Marsden, *Proc. R. Soc. A* 82 (1909) 495: ≈1 in 8,000 α reflected, platinum. Source: [chemteam.info](https://www.chemteam.info/Chem-History/GM-1909.html). (Two or more sources agree, corrected (platinum))
- Rutherford 1911 (*Phil. Mag.* 21, 669) "central charge"; "nucleus" from 1912. Source: Etymonline; ChemTeam. (Checked)
- Thomson's 1904 model (positive charge spread through the atom). Source: arXiv 1208.5262. (Checked (standard))
- 15-inch shell remark, lecture "Forty Years of Physics", 1936, printed 1938; wording varies †. Source: Needham & Pagel (eds), *Background to Modern Science* (CUP 1938). († (dagger in text))
- Gold leaf ≈ 0.1 µm (Rutherford's foils ≈ 0.4 µm) = 350–1,400 atoms. Source: Calculated for the book. (Checked / computed)
- Atom/nucleus diameter: gold ≈ 21,000, hydrogen ≈ 63,000; pea 8 mm → 165 m, 500 m; nuclear density 2.3×10¹⁷ kg/m³; 8 billion × 50 kg → 1.7 cm³. Source: Calculated for the book. (Calculated (r = 1.2 fm A^⅓; proton 0.8414 fm; illustrative 50 kg average))

#### 59. An old atom

- Becquerel 1896; Curie "radioactivité" 1898. Source: AIP exhibit; Britannica. (Two or more sources agree)
- Rutherford 1900 (*Phil. Mag.* 49): thorium emanation halves in about a minute. Source: Giunta, Le Moyne: [web.lemoyne.edu](https://web.lemoyne.edu/giunta/classicalcs/rutherhalfnote.html). (Checked)
- Libby radiocarbon 1949; Nobel Chemistry 1960. Source: nobelprize.org. (Two or more sources agree)
- Rn-222 half-life 3.82 d (text: 3.8). Source: NNDC. (Checked (standard))
- C-14 half-life 5,700 ± 30 yr. Source: Kutschera, *Radiocarbon* (2019). (Checked)
- Memoryless decay. Source: standard physics. (Checked)
- 3 × 5,700 = 17,100 years. Source: Calculated for the book. (Calculated)

Health check:

- Radon-222 half-life 3.8 days. Source: NNDC/IAEA (3.8235 d). (Confirmed)
- No health claim about radon is made. Source: (Radon is the second cause of lung cancer in the UK, UKHSA, [ukradon.org](https://www.ukradon.org/) — not needed on this page). (Confirmed)
- C-14 half-life ~5,700 years; Libby 1949, Nobel 1960. Source: Standard. (Confirmed)

#### 60. The bridge in the wind

- Opened 1 July 1940; collapsed 7 Nov 1940; wind 38 rising to 42 mph (68 km/h); "Galloping Gertie"; only death a dog (Tubby). Source: HistoryLink 5048; WSDOT; STRUCTURE magazine (Griggs 2022). (Two or more sources agree)
- Twisting from about 10 a.m.; collapse about 11 a.m. ("swayed for an hour"). Source: HistoryLink; WSDOT. (Checked)
- Torsional flutter (self-excited), not forced resonance: twisting ≈ 0.2 Hz vs vortex shedding ≈ 1 Hz at 42 mph. Source: Billah & Scanlan, *Am. J. Phys.* 59 (1991) 118; arXiv physics/0408101. (Two or more sources agree)
- Broughton Suspension Bridge, 12 April 1831, about 70 soldiers (60th Rifles) marching in step; faulty bolt also blamed. Source: Grace's Guide. (Checked, corrected (number))
- Army "break step" order †. Source: Grace's Guide (no primary order found). († (dagger in text))

#### 61. Down the plughole

- Coriolis 1835, *J. École polytechnique* 15. Source: Persson, *BAMS* 1998. (Checked (standard))
- Shapiro 1962 (*Nature* 196, 1080): tank about 6 ft (2 m) across, left ~24 h, anticlockwise. Source: Naked Scientists; Australian Geographic. (Two or more sources agree)
- Trefethen et al. 1965 (*Nature* 207, 1084), Sydney, clockwise. Source: Australian Geographic. (Checked)
- Ordinary baths: leftover motion and shape decide. Source: as above; Snopes. (Two or more sources agree)
- Equator demonstrations are tricks. Source: Snopes; Metabunk. (Two or more sources agree (sceptic and travel sources))
- Tropical cyclones rarely form within ≈500 km of the equator; turning senses. Source: NOAA HRD FAQ A15, D3. (Checked)
- London (51.5° N): Coriolis parameter f = 2Ω sin φ → inertial circle once per 15.3 h (Foucault period 30.6 h, not used after the AI adversarial review); swirl 1 cm/s at 10 cm → 63 s per turn, ≈ 880 times faster. Source: Calculated for the book. (Calculated, corrected)

#### 62. Red light, zinc plate

- Hallwachs 1888 (UV discharges negative zinc); Lenard 1902 (electron energy depends on frequency, not intensity). Source: Britannica "Photoelectric effect". (Two or more sources agree)
- Planck 14 Dec 1900; Einstein 1905 (*Ann. Phys.* 17, 132); Nobel 1921 (awarded 1922) "especially for … the photoelectric effect". Source: nobelprize.org. (Checked)
- "Photon" popularised by Lewis 1926 (earlier uses Troland 1916, Wolfers 1926). Source: Kragh, arXiv 1401.0293. (Corrected ("popularised"))
- Zinc work function ≈ 4.3 eV (3.7–4.9 by surface). Source: HyperPhysics. (Checked)
- Sunburn mainly UVB; B&W paper red-insensitive. Source: WHO; Ilford. (Checked (standard))
- Photon energies: 700 nm 1.77 eV, 550 nm 2.25, 450 nm 2.76, 300 nm 4.13, 254 nm 4.88. Source: Calculated for the book. (Calculated (hc/λe, CODATA exact))

#### 63. Sunlight in, heat out

- Fourier 1824 (*Ann. Chim. Phys.* 27). Source: Fleming, *Historical Perspectives on Climate Change*. (Checked (standard))
- Foote 1856 (*Am. J. Sci.* 22, 382), CO₂ cylinder hotter in sunlight; read by Henry at AAAS, 23 Aug 1856. Source: Physics Today; McNeill 2016. (Two or more sources agree)
- Tyndall 1859: water vapour and CO₂ strong absorbers, N₂ and O₂ nearly transparent (*Phil. Trans.* 151, 1861). Source: Royal Society. (Checked)
- Arrhenius 1896 (*Phil. Mag.* 41). Source: standard. (Checked)
- Greenhouses warm mainly by suppressing convection (Wood 1909, still debated). Source: Wood, *Phil. Mag.* 17 (1909) 319. (Checked (worded "mainly"))
- Effective temperature 254.6 K (−18.6 °C) for S = 1361 W/m², albedo 0.30 (−17.7 °C at 0.29); surface ≈ 288 K (15 °C); difference ≈ 33 °C. Source: Calculated for the book. (Calculated; checked for inputs (Kopp & Lean 2011; NASA fact sheet))
- Interior heat 47 TW (Davies & Davies 2010) → 0.092 W/m²; absorbed sunlight 238 W/m²; ratio ≈ 2,600. Source: Calculated for the book. (Calculated)
- N₂ + O₂ ≈ 99% of dry air. Source: NOAA. (Checked (standard: 78.08% + 20.95%))

#### 64. Floating in orbit

- Newton's cannonball in *Principia* (Def. V) and *A Treatise of the System of the World* (1728, figure). Source: Reed College Principia text; Wikipedia "Newton's cannonball". (Two or more sources agree)
- ISS 400–420 km, 7.66 km/s, ≈15.5 orbits a day (NASA rounds to 16). Source: NASA; issinfo.net. (Checked)
- NASA: astronauts float because in free fall; ~90% gravity. Source: NASA "What is microgravity?". (Checked)
- Parabolic flights 20–25 s. Source: Wikipedia "Reduced-gravity aircraft" (citing NASA). (Two or more sources agree)
- (6,400/6,800)² = 0.886; 6.25% farther; at 400 km (R = 6,371) 0.885; v(410 km) 7.67 km/s, 92.6 min, 15.5/day; surface orbit 7.91 km/s = 28,500 km/h. Source: Calculated for the book. (Calculated)

### D. Chemistry (puzzles 65–73)

#### 65. The log in the glass box

- Lavoisier, *Traité élémentaire de chimie* (1789), Part I ch. XIII: nothing is created in the operations of art or nature; equal quantity of matter before and after. Source: Gutenberg #52489 (French 1789), #30775 (Kerr's English 1790). (Checked)
- Lomonosov stated a conservation principle in a letter of 1748; little influence. Source: Shiltsev, arXiv 1801.00909. (Checked, corrected (framing))
- Van Helmont: 5 lb willow, 200 lb dried earth, 5 years, about 169 lb, earth lost about 2 oz; concluded water; printed 1648. Source: Howe 1965 translation, faculty.csbsju.edu/SSAUPE/biol115/van_helmont.htm. (Checked)
- Dry wood ≈ 50% C, 6% H, 44% O; heat of combustion ≈ 18–21 MJ/kg (19 used). Source: Rowell, *Handbook of Wood Chemistry*; BioResources hardwood study. (Checked (approximate))
- 10 kg wood → ≈ 13.7 kg O₂ used, ≈ 23.7 kg CO₂ + H₂O; heat 190 MJ ↔ 2.1 µg. Source: Calculated for the book. (Calculated (illustrative composition))
- Box reading 152 kg. Source: Calculated for the book. (Illustrative, labelled)

#### 66. Burning metal

- Phlogiston named and popularised by Stahl, early 1700s. Source: Wikipedia "Phlogiston theory"; Universalis. (Checked, corrected (Becher's *terra pinguis*, 1669, not used))
- Scheele: fire air by about 1772; book 1777. Source: Uppsala University Scheele page; ABAA listing. (Checked)
- Priestley: 1 Aug 1774, mercurius calcinatus, burning lens; "dephlogisticated air"; told Lavoisier in Paris, October 1774. Source: Science History Institute, Priestley biography. (Checked)
- Lavoisier: combustion as combination with oxygen; name from Greek "acid-former"; *oxygène* fixed 1787; all acids contain it (wrong). Source: Science History Institute, Lavoisier biography; chemteam.info. (Checked, corrected (date))
- 10 g Fe → 13.8 g Fe₃O₄ (14.3 g Fe₂O₃). Source: Calculated for the book. (Calculated)
- Candle under a jar goes out before all oxygen is used. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))

#### 67. The next metal down

- Melting points Li 180.5, Na 97.8, K 63.5, Rb 39.3, Cs 28.5 °C. Source: CRC; Wikipedia "Alkali metal". (Checked (±0.1 °corrected between sources))
- Steps 82.7, 34.3, 24.2, then 10.8. Source: Calculated for the book. (Calculated)
- Mendeleev's table March 1869 (17 Feb O.S.); detailed predictions 1871. Source: Scerri. (Checked (standard))
- Gallium 1875, scandium 1879, germanium 1886. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))
- Lecoq's first density 4.7; Mendeleev ≈ 5.9; 5.935 on remeasurement in 1876. Source: Wikipedia "Gallium"; *Scientific American* 15 June 1878. (Two or more sources agree)
- Lothar Meyer independent; Davy Medal shared 1882. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))

#### 68. Fruit or factory

- Ascorbic acid C₆H₈O₆; synthetic and natural identical. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))
- Synthetic vitamin E (all-rac) is a mixture of 8 stereoisomers; natural is RRR. Source: Linus Pauling Institute, vitamin E. (Checked)
- Natural vs synthetic vanillin told apart by isotope ratios (δ¹³C, SNIF-NMR). Source: [PMC8625575](https://pmc.ncbi.nlm.nih.gov/articles/PMC8625575/). (Checked)
- Wöhler 1828, letter to Berzelius (22 Feb 1828), "without needing kidneys or any animal". Source: Encyclopedia.com; *Ann. Phys. Chem.* 88 (1828). (Checked)
- "Killed vitalism" is a myth. Source: McKie, *Nature* 153 (1944) 608; Ramberg, *Ambix* 47 (2000) 170. (Two or more sources agree)
- Caffeine C₈H₁₀N₄O₂. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))

Health check:

- Natural and synthetic ascorbic acid are the same molecule and behave the same in the body. Source: Carr & Vissers, "Synthetic or food-derived vitamin C — are they equally bioavailable?", Nutrients 2013, [pmc.ncbi.nlm.nih.gov](https://pmc.ncbi.nlm.nih.gov/articles/PMC3847730/) (no difference in humans). (Confirmed)
- Option B (synthetic causes more side effects) is wrong. Source: Same. (Confirmed)
- Synthetic vitamin E is a mixture of eight stereoisomers, only one the main natural (RRR) form; not equivalent mg for mg. Source: Linus Pauling Institute, Vitamin E, [lpi.oregonstate.edu](https://lpi.oregonstate.edu/mic/vitamins/vitamin-E) ; NIH ODS fact sheet (1 mg all-rac ≈ 0.45 mg RRR), [ods.od.nih.gov](https://ods.od.nih.gov/factsheets/VitaminE-HealthProfessional/). (Confirmed)
- "an orange is far more than its vitamin C". Source: Carr & Vissers 2013. (Confirmed)
- Nature makes some of the most poisonous substances known. Source: Botulinum toxin (CDC). (Confirmed)
- Caffeine C8H10N4O2, same molecule in tea and cola. Source: PubChem CID 2519. (Confirmed)

#### 69. Drinking water

- Paracelsus, third of the *Seven Defences*, written 1538, printed 1564: "alle ding sind gifft / und nichts ohn gifft / Allein die dosis macht das ein ding kein gifft ist". Source: Aronson, BMJ blog, 9 Aug 2019. (Checked, corrected (printing date))
- Kidneys excrete at most ≈ 0.8–1.0 L of free water an hour. Source: systematic review, [PMC8663108](https://pmc.ncbi.nlm.nih.gov/articles/PMC8663108/). (Checked)
- Water-intoxication deaths in contests (e.g. Sacramento, 2007, ≈ 7.5 L in ~3 h) and in endurance runners (13% of 2002 Boston runners hyponatraemic; Almond et al., *NEJM* 2005). Source: ABC News; CBS; *NEJM* 352:1550. (Two or more sources agree)
- Botulinum toxin used medically in tiny measured doses (< 1 ng active toxin per 100-unit vial; mass not printed after the AI adversarial review, as it is product-specific). Source: [PMC3586120](https://pmc.ncbi.nlm.nih.gov/articles/PMC3586120/); *Toxins* 10:535. (Checked, corrected)
- Oxygen toxicity (lungs, nervous system). Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))

Health check:

- "drinking when thirsty is safe for healthy people". Source: NHS "Water, drinks and your health" (6–8 glasses a day; drink more in heat/exercise), [nhs.uk](https://www.nhs.uk/live-well/eat-well/food-guidelines-and-food-labels/water-drinks-nutrition/). (Confirmed)
- Kidneys excrete free water up to about a litre an hour. Source: Noakes et al., "Peak rates of diuresis in healthy humans during oral fluid overload", S Afr Med J 2001 (~900 mL/h), [pubmed.ncbi.nlm.nih.gov](https://pubmed.ncbi.nlm.nih.gov/11732457/) ; review [PMC8663108](https://pmc.ncbi.nlm.nih.gov/articles/PMC8663108/). (Confirmed)
- Hormones in hard exercise, stress or illness hold water back (ADH/vasopressin). Source: Hew-Butler et al., EAH consensus statement, Clin J Sport Med 2015. (Confirmed)
- Dilution of blood sodium, cell swelling, brain swelling: confusion, fits, death; "hyponatraemia". Source: NHS Inform / Merck Manual "Hyponatremia"; Hew-Butler 2015. (Confirmed)
- Deaths in water-drinking contests and in runners drinking more than they sweat. Source: Sacramento 2007 contest; Almond et al., NEJM 2005;352:1550. (Confirmed)
- "Confusion, vomiting, severe headache or fits after drinking a great deal need urgent medical help". Source: NHS 111/999 guidance for seizures and confusion. (Confirmed)
- Botulinum toxin injected in tiny doses into muscles to relax them. Source: NHS "Botulinum toxin" / MHRA product information. (Confirmed)
- "for many poisons there are doses too small to do measurable harm". Source: WHO IPCS principles of risk assessment. (Confirmed (hedged correctly; not claimed for all, e.g. genotoxic carcinogens))
- Try this: pure oxygen for long periods or at high pressure damages lungs and nervous system; divers and hospitals keep limits. Source: Standard (pulmonary and CNS oxygen toxicity); BTS oxygen guideline 2017 (target saturations), [brit-thoracic.org.uk](https://www.brit-thoracic.org.uk/quality-improvement/guidelines/emergency-oxygen/). (Confirmed)

#### 70. Lemon and coffee

- Lemon juice ≈ pH 2 (2–2.6), black coffee ≈ pH 5. Source: standard. (Checked (typical; labelled))
- 10³ = 1,000; 10⁴ = 10,000. Source: Calculated for the book. (Calculated)
- Ocean surface ≈ 8.2 → 8.1 since pre-industrial times; 10^0.1 = 1.26, "about a quarter more" (NOAA rounds to 30%). Source: NOAA ocean acidification page. (Checked, computed)
- Sørensen 1909, Carlsberg Laboratory; p_H, "hydrogen-ion exponent"; meaning of p unknown. Source: Wikipedia "pH"; Nørby, *TIBS* 2000. (Checked)

#### 71. The black powder

- 2 H₂O₂ → 2 H₂O + O₂; MnO₂ recovered unchanged. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))
- Kirchhoff 1811, acid converts starch to sugar, acid recovered. Source: *Educación Química* history of catalysis. (Checked)
- Döbereiner's lamp 1823. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))
- Berzelius 1835, "catalysis", Greek "loosening". Source: *Educación Química*. (Checked)
- Converters on US cars from the 1975 model year; lead poisons them. Source: Wikipedia "Catalytic converter"; Science History Institute. (Checked)
- Catalyst does not shift equilibrium. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))

Health check:

- Enzymes, most of them proteins, speed reactions, often millions of times over. Source: Standard biochemistry. Ribozymes are the exception, and "most" covers them.. (Confirmed)
- Enzymes in biological washing powders break down protein and fat stains in a cool wash. Source: Proteases and lipases are standard in detergents (standard).. (Confirmed)
- Converter turns CO and unburnt fuel into CO2 and water. Source: Not a health claim. It leaves out the reduction of NOx, which is acceptable here (main source table).. (Confirmed)
- Lead poisons converters; lead's harm to health was the other great reason for the phase-out. Source: main source table (Science History Institute); US EPA phase-down was health-driven.. (Confirmed)

#### 72. The bottom of the lake

- Max density 0.99997 g/cm³ at 3.98 °C; ice 0.9167. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard, CRC))
- Ice ≈ 9% larger; 10.6% of ice above sea water (1.025). Source: Calculated for the book. (Calculated)
- Bottom water under winter ice ≈ 4 °C, often 1–4 °C. Source: limnology standard. (Checked (with caveat in text))
- Pressure lowers freezing point ≈ 0.0074 °C/atm → 0.02 °C at 30 m. Source: Calculated for the book. (Calculated)
- Galileo 1612, *Discorso*, ice lighter than water. Source: Gutenberg #37729. (Checked)
- Florentine observers 1660s (Accademia del Cimento) saw water expand before freezing. Source: Standard reference fact or general knowledge; no single source recorded. (Likely (standard))
- Hope 1805, *Trans. RSE* 5, maximum density ≈ 39–40 °F. Source: Cambridge Core abstract. (Checked)

#### 73. A spoonful and the sea

- Teaspoon 1.67 × 10²³ molecules; oceans 2.66 × 10²³ teaspoons; ratio 1.6. Source: Calculated for the book. (Calculated)
- Ocean volume 1.332 × 10⁹ km³. Source: Charette & Smith, *Oceanography* 23(2) 2010. (Checked)
- Avogadro 1811; Loschmidt 1865; Perrin 1909 (name), Nobel 1926; Ostwald "Mol" 1890s; N_A fixed 20 May 2019. Source: Wikipedia "Avogadro constant", "Jean Perrin"; chemteam.info. (Checked, corrected (Ostwald date))
- Breath 0.5 L ÷ 24 L/mol × N_A ≈ 1.3 × 10²². Source: Calculated for the book. (Calculated)

### E. Life and health (puzzles 74–86)

#### 74. The drug that stopped working

- Darwin and Wallace papers read together, neither present, Linnean Society, 1 July 1858. Source: Linnean Society, 160th-anniversary note (2018): [linnean.org](https://www.linnean.org/the-society/news/2018/07/01/1st-july-2018-160th-anniversary-of-the-presentation-of-on-the-tendency-of-species-to-form-varieties) ; Wikipedia article on the joint paper (J. Proc. Linn. Soc. Zool. 3, 20 Aug 1858, 45–62). (Two or more sources agree)
- *Origin of Species* published November 1859. Source: standard (24 Nov 1859). (Checked)
- Darwin chose "natural selection" to mark its relation to breeders' selection. Source: Darwin, *Origin* (1859) ch. 3: "…in order to mark its relation to man's power of selection.". (Checked)
- Spencer coined "survival of the fittest" in 1864 (*Principles of Biology*). Source: Britannica, "survival of the fittest"; Darwin Correspondence Project, "Survival of the fittest": [darwinproject.ac.uk](https://www.darwinproject.ac.uk/commentary/survival-fittest). (Two or more sources agree)
- Wallace urged Darwin to adopt it (letter of 2 July 1866); Darwin added it in the 5th edition of the *Origin*, 1869. Source: Darwin Correspondence Project (above); phrases.org.uk / History Revealed (5th ed., 10 Feb 1869). The text does not mention the 1868 *Variation* use, which one source claims and we did not verify.. (Two or more sources agree)
- Weeds resistant to weedkillers; lice resistant to lotions; resistance often has a fitness cost. Source: See the health check below.. (Checked in the health check for this puzzle)
- 10^10 × 10^-8 = 100; 100 × 2^20 ≈ 1.05 × 10^8 (illustrative). Source: Calculated for the book. (Calculated)

Health check:

- Resistance belongs to the bacteria, not the patient (A wrong). Source: WHO fact sheet "Antimicrobial resistance": AMR occurs "when bacteria, viruses, fungi and parasites do not respond to these medicines", [who.int](https://www.who.int/news-room/fact-sheets/detail/antimicrobial-resistance). (Confirmed)
- Variants arise by chance copying errors or genes picked up from other bacteria, and may be present before treatment or arise during it. Source: Mutation and horizontal gene transfer (standard; WHO, above).. (Confirmed)
- 10^10 × 10^-8 = 100 survivors; 20 half-hour doublings → ~100 million. Source: The arithmetic is correct and the illustration is labelled. However, it implies the 100 survivors regrow unopposed. In a patient, the immune system usually clears small numbers, and that is why most treated infections do not relapse. A clinician would want "unchecked" added.. (Text revised after this check)
- Weeds resistant to weedkillers; lice resistant to lotions once used against them. Source: Permethrin and malathion resistance in UK head lice is documented (UK schoolchild study: permethrin 87% and malathion 64% failure, cited via NHS Scotland formulary, [formulary.nhs.scot](https://formulary.nhs.scot/east/skin/parasite-infections/head-lice/) ).. (Confirmed)
- In a gut with no drug, a resistant bacterium often grows more slowly. Source: Andersson & Hughes, *Nat Rev Microbiol* 8 (2010) 260, [doi.org](https://doi.org/10.1038/nrmicro2319) . Most resistance carries a cost, which can be compensated; "often" is the right word.. (Confirmed)

#### 75. Soho, 1854

- "Upwards of five hundred fatal attacks of cholera in ten days" (616 in all). Source: Snow, *On the Mode of Communication of Cholera*, 2nd ed. 1855 (Gutenberg #72894). (Checked)
- Workhouse: 535 inmates, 5 deaths, own well; brewery: above 70 workmen, none died, malt liquor and own well. Source: Snow 1855. (Checked)
- Hampstead widow had Broad Street water brought to her and died (2 Sept); unnamed in Snow. Source: Snow 1855. (Checked)
- Contamination from a nearby cesspool. Source: Wikipedia "1854 Broad Street cholera outbreak". (Checked (standard; Whitehead's investigation))
- Handle removed 8 Sept after Snow met the Guardians on 7 Sept; deaths already falling (Snow's own remark). Source: Snow 1855. (Checked)
- Vienna 1846: First Clinic 459/4,010 = 11.4%; Second 105/3,754 = 2.8%; alternate-day admission; chlorinated lime from mid-May 1847; June 2.2%, July 1.2%, Aug 1.9%. Source: Wikipedia "Ignaz Semmelweis"; Loudon, *JRSM* 2013. (Two or more sources agree)
- Semmelweis died 13 Aug 1865 in a Vienna asylum of pyaemia from a hand wound; beating disputed; legends dagger-labelled. Source: Wikipedia; kepmas.hu. (Checked, corrected)
- Pacini 1854; Koch pure culture Jan 1884, anthrax 1876, TB 24 March 1882; Pasteur 1861–62. Source: Wikipedia "Filippo Pacini", "Vibrio cholerae", "Louis Pasteur"; PBS. (Checked)

Health check:

- Cholera caused by a germ in pump water from a nearby cesspool. Source: WHO cholera fact sheet: caused by "consuming food or water contaminated with the bacterium *Vibrio cholerae*", [who.int](https://www.who.int/news-room/fact-sheets/detail/cholera) ; Snow 1855 (main source table). (Confirmed)
- Many diseases are caused by microbes passing person to person in water, food, air or by touch; blocking routes can reduce or stop spread. Source: Std. "Many" allows for vector-borne exceptions.. (Confirmed)
- Handwashing is a central reason. Source: Std (WHO hand hygiene).. (Confirmed)
- Vienna 1846: 11.4% vs 2.8%; chlorinated lime May 1847; ~2% within months. Source: Loudon, *JRSM* 2013 (main source table). (Confirmed)
- Semmelweis died in an asylum in 1865 of an infected wound. Source: main source table. (Confirmed)

#### 76. How many must be immune?

- Threshold 1 − 1/R₀ under random mixing and complete immunity; 1 − 1/15 = 93.3%; 15 × 0.2 = 3; 15 × 0.05 = 0.75; R₀ = 2 → 50%. Source: Fine, Eames & Heymann, *Clin Infect Dis* 52 (2011) 911. (Checked; computed)
- Measles R₀ often quoted 12–18, real estimates vary more. Source: Guerra et al., *Lancet Infect Dis* 17 (2017) e420. (Checked)
- WHO: ≥ 95% coverage with two doses. Source: WHO/UNICEF statement. (Checked)
- Two MMR doses ≈ 97% effective against measles (one dose 93%). Source: CDC measles vaccine considerations. (Checked)
- Jenner 14 May 1796, James Phipps (8), cowpox from Sarah Nelmes; challenged 1 July 1796. Source: historyofvaccines.org; Wikipedia "James Phipps". (Checked)
- *vacca*; smallpox declared eradicated 8 May 1980 (WHA33.3). Source: Science Friday; UN press release. (Checked)

Health check:

- Threshold 1 − 1/R0; 93.3% for R0 = 15; it does not mean no cases. Source: Fine, Eames & Heymann, *Clin Infect Dis* 52 (2011) 911 (main source table). The idealisation is labelled on the page.. (Confirmed)
- Vaccination protects babies too young and people with medical contraindications. Source: Std (NHS/UKHSA herd protection messaging).. (Confirmed)
- WHO aims for ≥95% coverage with each of two doses. Source: WHO/UNICEF (main source table). (Confirmed)
- "two doses protect about 97% of those who get them". Source: CDC: two doses "97% (range: 67% to 100%) effective", [cdc.gov](https://www.cdc.gov/measles/hcp/vaccine-considerations/index.html) . NHS: after 2 doses "around 99% of people will be protected against measles", [nhs.uk](https://www.nhs.uk/vaccinations/mmr-vaccine/) . A UK reader who checks the NHS page will see 99%, so "97 to 99%" is safer.. (Text revised after this check)
- Jenner 1796, Phipps, Nelmes; smallpox eradicated 1980. Source: main source table. (Confirmed)

#### 77. The colour that came back

- Mendel: Brünn society, 8 Feb and 8 Mar 1865; published 1866; flower colour 705 : 224 (3.15 : 1). Source: MendelWeb translation; Villanova. (Checked; computed)
- Rediscovery 1900 chiefly by de Vries and Correns. Source: Villanova. (Checked)
- Johannsen "gene" 1909. Source: [PMC4258773](https://pmc.ncbi.nlm.nih.gov/articles/PMC4258773/). (Checked)
- Fisher's "too good" anomaly unresolved; fraud unproven. Source: Franklin et al., *Ending the Mendel–Fisher Controversy* (2008). (Checked)
- Eye colour polygenic; brown-eyed parents can have blue-eyed children. Source: The Tech "Ask a geneticist"; Cosmos. (Checked)
- Try: 2/3 of purple plants carry p. Source: Calculated for the book. (Calculated)

#### 78. Counting the letters

- Human DNA ≈ A 30, T 30, G 20, C 20 (Chargaff 1952: 29.3/30.0/20.7/20.0); Chargaff 1950. Source: Wikipedia "Chargaff's rules" (citing Bansal 2003). (Checked)
- Photo 51, May 1952, Gosling under Franklin. Source: Embryo Project (ASU). (Checked)
- Wilkins showed it to Watson in late January 1953 without Franklin's knowledge; MRC report (Dec 1952) passed by Perutz. Source: historyofinformation.com; *Nature* 424, 127. (Checked)
- *Nature* 171, 25 April 1953: three papers. Source: University of Manchester news. (Checked)
- Cobb & Comfort, *Nature* 616 (2023) 657: "equal member". Source: as above. (Checked)
- Franklin died 16 April 1958, aged 37; never nominated; 1962 Nobel (Physiology or Medicine) to Watson, Crick, Wilkins. Source: CSHL; LMU. (Checked, corrected)

Health check:

- Human DNA about 30% A, 20% G. Source: Chargaff 1952 (29.3/20.7) (main source table). (Confirmed)
- Base pairing lets laboratories read DNA and compare it in paternity and criminal cases. Source: Std. Profiling uses STR/PCR, which relies on primer pairing.. (Confirmed)
- Franklin died 1958 aged 37. Source: main source table. (Confirmed)

#### 79. Not made of cells

- Hooke, *Micrographia* (1665), Obs. XVIII: "much like a Honey-comb", "these pores, or cells". Source: Gutenberg text via MIT OCW reading. (Checked)
- Monks' cells naming †. Source: as above. († (dagger))
- Schleiden 1838, Schwann 1839, Virchow 1855 drawing on Remak. Source: standard; Virchow's *Archiv* 1855. (Checked)
- Influenza virion ≈ 80–120 nm; red cell 7.5–8 µm → "some seventy times narrower"; 125–133 cells per mm. Source: Calculated for the book. (Checked (standard); computed)
- ≈ 30 trillion human cells; ≈ 200 cell types. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (Sender et al. 2016); standard)

Health check:

- Influenza virus about 0.1 µm, ~70× narrower than a red cell. Source: Frontiers in Microbiology review, "80–120 nm", envelope "derived from the host cellular membrane during the budding", [frontiersin.org](https://www.frontiersin.org/journals/microbiology/articles/10.3389/fmicb.2011.00269/full). (Confirmed)
- "a few strands of genetic material, RNA". Source: Influenza A and B have exactly eight negative-sense RNA segments ([PMC3074182](https://pmc.ncbi.nlm.nih.gov/articles/PMC3074182/), [pmc.ncbi.nlm.nih.gov](https://pmc.ncbi.nlm.nih.gov/articles/PMC3074182) ). "A few" undersells eight; the exact number costs nothing.. (Text revised after this check)
- Fatty envelope taken from host cell; no machinery to make proteins or harness energy; multiplies only in a cell. Source: Same review. The virus carries its own RNA polymerase, but that is not protein-making or energy machinery.. (Confirmed)
- Typical adult ~30 million million cells. Source: Sender et al. 2016 (main source table). Hatton et al. 2023 give ~36 trillion for a man and ~28 trillion for a woman, consistent with this.. (Confirmed)
- "of roughly two hundred kinds". Source: ~200 is the traditional textbook count. Hatton et al., *PNAS* 120 (2023) e2303077120, [ncbi.nlm.nih.gov](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10523466/) , work with ~400 cell types, and cell atlases split further still. "a few hundred kinds" is safe.. (Text revised after this check)
- Nerve cells a metre long; red cells a few µm. Source: Std (sciatic motor axons).. (Confirmed)

#### 80. Eighty per cent

- Heritability = share of phenotypic variance in a population associated with genetic variance; not individual. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))
- Height heritability ≈ 0.8 or more in twin studies of well-fed populations. Source: Silventoinen et al., *Twin Res.* 6 (2003) 399. (Checked)
- Heights rose in the 20th century (up to +20 cm, South Korean women). Source: NCD-RisC, *eLife* 2016. (Checked)
- Fisher 1918; Lush 1937. Source: *Trans. R. Soc. Edinb.* 52; Springer *Biol. Theory* 2021. (Checked, corrected (Lush year))
- Finger number: genetic but low heritability. Source: Standard fact; no single source recorded. (Checked (teaching example; unattributed))

Health check:

- Heritability is a population variance statistic, not individual; can change with environment; equalising environment can raise it. Source: Std quantitative genetics (Visscher, Hill & Wray, *Nat Rev Genet* 2008).. (Confirmed)
- Heights rose by several cm over the 20th century, in a few by much more. Source: NCD-RisC, *eLife* 2016 (main source table). (Confirmed)
- Twin studies of well-fed populations put height heritability at roughly 70–90%. Source: Silventoinen et al., *Twin Res* 6 (2003) 399 (main source table). (Confirmed)
- Finger number genetic but low heritability. Source: Std teaching example.. (Confirmed)

#### 81. How much of you is bacteria?

- 3.8 × 10¹³ bacteria, 3.0 × 10¹³ human cells (70 kg reference man); 0.2 kg bacteria; colon contents ≈ 0.4 kg; RBC 84% of cells; defecation removes ¼–⅓ of colon bacteria. Source: Sender, Fuchs & Milo, *PLOS Biol* 14 (2016) e1002533. (Checked)
- 10:1 from Luckey 1972 (1 L × 10¹¹/g = 10¹⁴ vs 10¹³). Source: as above. (Checked)
- Shares 56% by number, 0.29% by mass; 30/10 = 3, 100/38 = 2.6. Source: Calculated for the book. (Calculated)

Health check:

- 38 trillion bacteria vs 30 trillion human cells; ~0.2 kg; colon contents 400 g; RBC 84%; defecation removes ¼–⅓. Source: Sender, Fuchs & Milo, *PLOS Biol* 14 (2016) e1002533 (main source table). (Confirmed)
- No health advice given (no probiotic or diet claims). Source: —. (Confirmed)

#### 82. How much of your brain?

- Brain ≈ 2% of weight, ≈ 20% of resting energy. Source: Raichle & Gusnard, *PNAS* 99 (2002) 10237. (Checked)
- James 1907: "We are making use of only a small part of our possible mental and physical resources" (no number). Source: psychclassics.yorku.ca. (Checked)
- Lowell Thomas 1936 preface: "ten percent of his latent mental ability" †. Source: Wikipedia "Ten percent of the brain myth"; Beyerstein 1999. (Checked (quote), † (attribution to James))
- Einstein misattribution. Source: Beyerstein 1999. (Checked (no source exists))
- ≈ 86 billion neurons; glia:neuron < 1:1. Source: Azevedo et al. 2009; von Bartheld et al., *J Comp Neurol* 524 (2016) 3865. (Checked)

Health check:

- Brain 2% of weight, ~20% of resting energy; ~10× per gram. Source: Raichle & Gusnard, *PNAS* 99 (2002) 10237 (main source table). (Confirmed)
- Damage to most parts takes something away; some small injuries go unnoticed. Source: Std. Covert ("silent") strokes are common.. (Confirmed)
- "If every nerve cell fired together, the result would not be genius but a seizure". Source: The rhetoric works, but a seizure is a burst of abnormal, synchronised activity in large groups of neurons, not all of them firing at once. A neurologist would quibble; a small rewording fixes it.. (Text revised after this check)
- 86 billion neurons, similar number of other cells, mostly glia. Source: Azevedo et al. 2009; von Bartheld et al. 2016 (main source table). (Confirmed)

#### 83. Where does the blood go?

- 70 mL × 70/min = 4.9 L/min; 294 L/h; 7,056 L/day; exercise 20–25 L/min → 12–15 s. Source: Calculated for the book. (Calculated)
- Blood volume ≈ 5 L; red cell life ≈ 120 days. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))
- Galen; Ibn al-Nafīs (13th c.); Harvey *De motu cordis* 1628, ch. 9 argument with ranges; Malpighi 1661 capillaries. Source: Wikisource Harvey ch. 9; standard. (Checked)
- Harvey's ligature/vein-valve demonstrations. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard; *De motu* ch. 13))

Health check:

- Adult ~5 L blood; stroke volume ~70 mL; HR ~70; ~4.9 L/min; circuit ~once a minute. Source: Std physiology (Guyton & Hall). 5 L is the reference-man figure; women have ~4–4.5 L, and "about" covers this.. (Confirmed)
- Red blood cell lasts about four months. Source: ~120 days (standard). (Confirmed)
- Arteries out, veins back, carrying oxygen and nutrients, removing wastes. Source: Std. (Confirmed)
- Fit person in hard exercise 20–25 L/min. Source: Std. Elite endurance athletes reach 35–40 L/min, so the figure is conservative.. (Confirmed)
- Arm vein valves direct flow towards the heart; Harvey's demonstration. Source: Harvey *De motu* ch. 13 (main source table). (Confirmed)

#### 84. A plant in the dark

- de Mairan 1729, mimosa leaf movements persist in darkness; reported to the Académie. Source: Emery et al., *J Biol Rhythms* 38 (2023) 245; ScienceBlogs. (Checked, corrected ("Marchant" dropped))
- de Candolle 1832, ≈ 22 h in constant light. Source: as above. (Checked)
- Human free-running period 24.18 h. Source: Czeisler et al., *Science* 284 (1999) 2177. (Checked)
- Halberg "circadian" 1959. Source: as above. (Checked)
- 0.2 h × 10 = 2 h. Source: Calculated for the book. (Calculated)
- Answer diagram: illustrative period of about 23 h (mimosa periods reported ≈ 22–23 h), not the human 24.2 h. Source: de Candolle 1832. (Illustrative, labelled)

Health check:

- Cells in plants and in us run a gene–protein feedback loop of about a day; light resets it; it free-runs and may damp in constant conditions. Source: Std chronobiology. Plant and animal clock genes differ, but both use a transcription–translation loop.. (Confirmed)
- Human clock about 24.2 h; drift 2 h in 10 days. Source: Czeisler et al., *Science* 284 (1999) 2177 (main source table). (Confirmed)
- Jet lag: body keeps old time; days of daylight at the new place shift it. Source: NHS "Jet lag": "usually improves within a few days"; "natural light will help your body clock adjust", [nhs.uk](https://www.nhs.uk/conditions/jet-lag/). (Confirmed)

#### 85. Peanut or sugar?

- Sucrose 5,645 kJ/mol → 16.5 kJ/g; peanut (49.2% fat, 25.8% protein, 16.1% carbohydrate; 39.5/23.6/17.2 kJ/g) → 28.3 kJ/g; ratio 1.7; food tables 567 vs 387 kcal/100 g → 1.47. Source: Calculated for the book. (Checked (standard, USDA); computed)
- Protein: nitrogen leaves as urea. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))
- Lavoisier & Laplace, guinea pig, ice calorimeter, winter 1782–83; respiration a slow combustion. Source: IUPAC *Pure Appl. Chem.* 59 (1987) 91. (Checked)
- Atwater's respiration calorimeter, 1890s. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))
- kcal = 4.184 kJ. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (definition))

Health check:

- Fat ~39 kJ/g, sugar ~16.5 kJ/g; peanut ~28 kJ/g; ratio 1.7. Source: Gross combustion values (main source table, ). (Confirmed)
- Body gets nearly as much energy from digestible fat and carbohydrate as a flame would. Source: Atwater factors: 37 vs ~39 kJ/g for fat, 17 vs ~17 kJ/g for carbohydrate (standard).. (Confirmed)
- Protein: nitrogen leaves as urea, which still holds energy; tables allow for that and for fibre; peanuts ~1.5× sugar. Source: USDA 567 vs 387 kcal/100 g (main source table). Measured metabolisable energy of whole nuts can be lower still than the tables, but this is not needed on the page.. (Confirmed)
- Sugar "quick to use" but less energy per gram. Source: Std. (Confirmed)
- Label Calorie = kcal = 4.184 kJ. Source: Definition. (Confirmed)

#### 86. Take away the starfish

- Paine: from June 1963, Mukkaw Bay (near Neah Bay, Washington), *Pisaster ochraceus* removed; 15 → 8 species; *Balanus glandula* first, then *Mytilus californianus* and *Mitella polymerus*. Source: Paine, *Am. Nat.* 100 (1966) 65–75, p. 70. (Checked)
- "Keystone species", 1969. Source: Paine, *Am. Nat.* 103 (1969) 91. (Checked)
- Otters, urchins and kelp. Source: Estes & Palmisano, *Science* 185 (1974) 1058. (Checked)

### F. Earth and sky (puzzles 87–95)

#### 87. Measuring the world with a stick

- Eratosthenes of Cyrene, c. 276–194 BC, librarian at Alexandria. Source: MacTutor, "Eratosthenes": [mathshistory.st-andrews.ac.uk](https://mathshistory.st-andrews.ac.uk/Biographies/Eratosthenes/) ; Wikipedia "Eratosthenes". (Two or more sources agree)
- His book is lost; method survives in Cleomedes; 1/50 of a circle; 5,000 stades; 250,000 stades. Source: Cleomedes, *Caelestia* I.7.49–52 (Bowen & Todd trans., 2004), via Wikipedia "Eratosthenes" and Heath's translation quoted at roger-pearse.com (2017); MacTutor (250,000). (Two or more sources agree)
- Accuracy depends on the unknown length of the stade; 1% claims doubtful †. Source: MacTutor; Wikipedia lists stade values 160–210 m (Matthew 2023). 250,000 stades = 39,375 km (157.5 m) to 46,250 km (185 m), calculated. (Two or more sources agree)
- Credited with coining "geography". Source: Roller, *Eratosthenes' Geography* (2010), pp. ix, 1, via Wikipedia. (Checked (secondary))
- Aswan–Alexandria: latitude difference 7.11°, meridian distance ≈ 790 km, circumference ≈ 40,000 km; Alexandria ~3° west. Source: calculated (coordinates Aswan 24.089 N 32.900 E; Alexandria 31.200 N 29.919 E). (Calculated)
- Midsummer Sun about a third of a degree short of overhead at Aswan in Eratosthenes' day. Source: obliquity c. 240 BC ≈ 23.72° (IAU polynomial extrapolated; literature gives ≈ 23.7°) vs Aswan latitude 24.09° → ≈ 0.36°. (Calculated (approximate))
- Try this: 36 × 1,110 = 39,960 km. Source: Calculated for the book. (Calculated)

#### 88. Midwinter in London

- Perihelion early January, ≈147.1 million km; aphelion early July, ≈152.1 million km. Source: USNO "Earth's Seasons"; NASA Earth fact sheet. (Checked)
- London noon Sun ≈15° (Dec) and ≈62° (Jun); day under 8 h and over 16½ h; ≈7% more sunlight in January. Source: calculated (51.5° N, obliquity 23.44°, refraction allowance −0.833°). (Calculated)
- Ptolemy's obliquity 23°51′(20″). Source: Almagest I.12 (Toomer); Jones, *JHA* 33 (2002). (Checked)
- Hipparchus: spring 94½ d, summer 92½ d. Source: Almagest III.4 (Toomer p. 153). (Checked)
- Kepler 1609: Mars on an ellipse. Source: *Astronomia nova*; extended to all planets in the *Epitome* (1617–21). (Checked, corrected)
- Northern spring+summer ≈186 d, autumn+winter ≈179 d. Source: USNO equinox/solstice dates. (Checked)

#### 89. A thin crescent

- Parmenides "alien light". Source: Diels–Kranz 28 B14. (Checked)
- Anaxagoras: the Moon's light comes from the Sun. Source: Plato, *Cratylus* 409a. (Checked)
- Aristotle: curved shadow in lunar eclipses → spherical Earth. Source: *De Caelo* II.14, 297b. (Checked)
- Leonardo explained earthshine, c. 1506–1510. Source: Codex Leicester f. 2r; Louvre 2019 catalogue. (Checked)
- Lunar eclipses only at full Moon. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))

#### 90. High water

- Barycentre ≈4,700 km from the Earth's centre. Source: NASA (4,671 km); calculated (4,668 km). (Checked, computed)
- Sun's tide ≈46% of the Moon's ("a little under half"). Source: NOAA tides tutorial; calculated (0.46). (Checked, computed)
- High tides on average 12 h 25 min apart. Source: NOAA. (Checked)
- Gulf of Mexico diurnal tides. Source: NOAA "Types and causes of tidal cycles". (Checked)
- Galileo 1632 attributed tides to the Earth's motions; "the moon's dominion over the waters". Source: *Dialogue*, Fourth Day (Drake transl.); Drake's wording confirmed via Wikiquote, "Theory of tides", and Blåsjö, [arXiv:2102.06595](https://arxiv.org/abs/2102.06595), fn. 203; the page (p. 462 of the 1953 edition) is likely but was not seen. The book prints only the phrase. (Checked (phrase))
- Newton 1687 explained tides by Moon and Sun. Source: *Principia* I.66, III.24, 36–37. (Checked)

#### 91. An ocean opening

- 4,000 km at 3 cm a year → 133 million years. Source: Calculated for the book. (Calculated (illustrative, labelled))
- Plates move about as fast as fingernails grow. Source: USGS FAQ "How fast do tectonic plates move?". (Checked)
- South Atlantic opening ≈130–140 Ma; present full rate ≈3–3.5 cm/yr. Source: USGS event summaries; DeMets et al. 2010 (MORVEL). (Checked)
- Wegener lecture 6 Jan 1912, Frankfurt; *Mesosaurus*, glacial evidence. Source: Geological Society, "Pioneers of plate tectonics". (Checked)
- Forces too weak (Jeffreys 1924); Greenland ≈20 m/yr claim wrong. Source: Polar Record, "The supposed westward drift of Greenland". (Checked, corrected)
- Hess 1960–62 (preprint 1960); Dietz 1961, independently; Vine & Matthews 1963; Morley independently (rejected 1963); plate tectonics 1965–68. Source: GSA Buddington volume; *Nature* 199, 947; Dietz, *Nature* 190 (1961) 854. (Checked)
- Oldest open-ocean floor ≈180 Ma. Source: — (eastern Mediterranean may be older; text says "open oceans"). (Checked (standard))

#### 92. A join in the rocks

- Siccar Point basal breccia made of greywacke fragments. Source: BGS Earthwise, Siccar Point excursion; Scottish Geology Trust geoguide. (Checked)
- Lower beds ≈435 Ma (Llandovery); upper ≈370 Ma (Late Devonian); gap ≈65 Myr. Source: BGS Earthwise. (Checked, corrected)
- Ussher 1650, 4004 BC. Source: *Annales Veteris Testamenti*. (Checked)
- Steno 1669. Source: *Prodromus*. (Checked)
- Hutton read 1785, "we find no vestige of a beginning,—no prospect of an end.". Source: Trans. RSE 1 (1788) 304. (Checked (scan))
- 1788 voyage with Playfair and Hall. Source: Playfair, Trans. RSE 5 (1805) 71–73. (Checked (scan))
- "The mind seemed to grow giddy by looking so far into the abyss of time". Source: ibid., p. 73. (Checked (scan))
- Patterson 1956, 4.55 ± 0.07 Gyr; modern 4.54 ± 0.05. Source: *GCA* 10, 230; USGS. (Checked)
- Last 10,000 years ≈ 0.19 s of a day. Source: Calculated for the book. (Calculated)

#### 93. Stars without end

- Kepler 1610. Source: *Dissertatio cum Nuncio Sidereo*. (Checked)
- De Chéseaux 1744: every point as bright as the Sun's disc. Source: *Traité de la comète*, App. II; *Physics Today* "Why the sky is dark at night". (Checked, corrected)
- Olbers 1823, absorption. Source: Bode's *Jahrbuch für 1826*; Linda Hall Library. (Checked, corrected (date))
- Poe, *Eureka* (1848): light from the background has not yet reached us. Source: Gutenberg #32037. (Checked)
- Kelvin 1901. Source: *Phil. Mag.* 2 (1901) 161. (Checked)
- Finite age the main resolution; expansion minor. Source: Harrison, *Darkness at Night* (1987). (Checked)
- Dust would heat and re-radiate. Source: Standard reference fact or general knowledge; no single source recorded. (Checked (standard))
- Age 13.8 Gyr. Source: Planck 2018. (Checked)

#### 94. Raisins in the dough

- Raisins: 1 cm and 3 cm. Source: Calculated for the book. (Calculated)
- ≈70 km/s per Mpc (3.26 Mly); measurements 67–73. Source: Planck 2018; Riess et al. 2022. (Checked)
- Andromeda approaching. Source: van der Marel et al. 2012 (≈110 km/s relative to the Milky Way). (Checked)
- Slipher from 1912; 21 of 25 receding by 1917. Source: *Proc. Am. Phil. Soc.* 56 (1917). (Checked)
- Lemaître 1927 derived and estimated the rate (625 km/s/Mpc) from Slipher's velocities and Hubble's distances. Source: *Ann. Soc. Sci. Bruxelles* A47; Livio, *Nature* 479 (2011). (Checked)
- Hubble 1929. Source: *PNAS* 15, 168. (Checked)
- IAU 2018 recommendation "Hubble–Lemaître law". Source: IAU press release iau1812. (Checked)
- 1 Gly at 70 km/s/Mpc → ≈21,500 km/s; 1/H₀ ≈ 14.0 Gyr. Source: Calculated for the book. (Calculated)

#### 95. Two forecasts

- Predictability limit ≈2 weeks. Source: Zhang et al., *J. Atmos. Sci.* 76 (2019). (Checked)
- Daily temperature chart. Source: Calculated for the book. (Illustrative (invented, labelled))
- IMO 1935, 1901–1930 normals; WMO standard 1991–2020. Source: WMO-No. 1203. (Checked)
- Lorenz 1963; 1975 two kinds of predictability. Source: *J. Atmos. Sci.* 20; GARP Pub. 16. (Checked)
- Herbertson 1901: "Climate is what on an average we may expect, weather is what we actually get.". Source: *Outlines of Physiography* (1901), via Quote Investigator. (Checked)

### G. How we know (puzzles 96–100)

#### 96. Normal temperature

- 36.5–37.5 °C → 97.7–99.5 °F; 200 °C → 392 °F. Source: Calculated for the book. (Calculated)
- Mackowiak et al. 1992: 148 adults 18–40, mean 36.8 °C (98.2 °F) oral, daily amplitude ≈0.5 °C, lowest 6 a.m., highest 4–6 p.m.. Source: *JAMA* 268 (1992) 1578 (PMID 1302471). (Checked)
- Protsiv et al. 2020: decline ≈0.03 °C per birth decade. Source: *eLife* 9:e49555. (Checked)
- Wunderlich 1868; "some millions" of observations, nearly 25,000 cases; axilla; 15–20 min to settle. Source: 1871 New Sydenham Society translation (archive.org). (Checked)
- 1871 translation prints 98.6°. Source: ibid. §2. (Checked)
- His thermometer read almost 2 °C high. Source: Mackowiak, Freakonomics Radio no. 268; Mackowiak & Worden, *Clin Infect Dis* 18 (1994) 458. (Checked (oral account))

Health check:

- Normal temperature differs between people, varies through the day (lowest early morning, highest late afternoon), and depends on site. Source: Mackowiak et al., *JAMA* 268 (1992) 1578 (main source table); NHS "Fever in adults": "Normal body temperature is different for everyone and changes during the day", [nhs.uk](https://www.nhs.uk/conditions/fever-in-adults/). (Confirmed)
- 1992 study: 148 adults, mean 36.8 °C (98.2 °F) oral, ~0.5° daily swing. Source: Mackowiak 1992 (main source table). (Confirmed)
- 2020 study: average temperatures have fallen slightly since the 19th century. Source: Protsiv et al., *eLife* 9 (2020) e49555 (main source table). (Confirmed)
- Answer interval 97.7–99.5 °F is about rounding, not a clinical normal range. Source: The page never presents it as a clinical range, and gives no fever threshold. NHS uses 38 °C or above as a high temperature, so nothing on the page conflicts with that. No change needed.. (Confirmed)
- Wunderlich thermometer read nearly 2 °C high ("reportedly"). Source: Mackowiak & Worden 1994 (main source table). (Confirmed)

#### 97. Small pushes on the way to Mars

- 1 lbf·s = 4.448 N·s (exact definition 4.448 221 615 260 5). Source: Calculated for the book. (Checked, computed)
- MCO lost 23 Sept 1999; lbf-s instead of N-s; underestimated by 4.45; 226 km planned, 57 km, 80 km survivable. Source: MCO MIB Phase I report, 10 Nov 1999. (Checked)
- Gimli Glider 23 July 1983; 1.77 vs ≈0.8; less than half the fuel; no deaths. Source: Board of Inquiry (Lockwood 1985). (Checked)
- 50 mph = 80.5 km/h (61% more). Source: Calculated for the book. (Calculated)
- Metre 1791; Delambre & Méchain 1792–98; standards 1799; Metre Convention 1875; SI 1960. Source: BIPM; Alder, *The Measure of All Things*. (Checked)

#### 98. Two explanations

- 0.5² = 25%; 0.9⁵ ≈ 59%; Try: 0.8² = 64%. Source: Calculated for the book. (Calculated (illustrative, labelled))
- Ockham c. 1287–1347; "Numquam ponenda est pluralitas sine necessitate". Source: SEP "William of Ockham"; *In I Sent.* d. 27 q. 2 (Thorburn appendix). (Checked)
- "Entities must not be multiplied…" not in Ockham; Punch 1639 (*sine*); Clauberg 1654 (*praeter*). Source: Thorburn, *Mind* 27 (1918) 345. (Checked, corrected)
- Froidmont 1649 *novacula Occami*. Source: Ariew (via secondary). (Checked (likely))
- Principle older (Aristotle). Source: SEP "Simplicity". (Checked)

#### 99. Both came true

- *Logik der Forschung* 1934 (imprint 1935); English 1959. Source: Standard fact; no single source recorded. (Checked)
- 1919: Eddington; Adler anecdote, "thousandfold experience" / "thousand-and-one-fold". Source: *Conjectures and Refutations* (1963), ch. 1. (Checked)
- Neptune 1846 near Le Verrier's position; Vulcan never found; GR explains Mercury. Source: Standard fact; no single source recorded. (Checked)
- Duhem–Quine point (blame an auxiliary assumption). Source: SEP "Karl Popper" §4. (Checked)

#### 100. A whisper and a cough

- Over ×100 after 49 trips (1.1⁴⁹ ≈ 107), over ×1,000 after 73 (≈1,051); 49 trips ≈ 0.49 s at ≈0.01 s a trip. Source: Calculated for the book. (Calculated (illustrative, labelled))
- Ktesibios float regulator. Source: Mayr, *The Origins of Feedback Control* (1970). (Checked)
- Watt's governor 1788; Boulton's letter on mill governors (28 May 1788). Source: Mayr 1970. (Checked)
- Maxwell, "On Governors", *Proc. R. Soc.* 16 (1868). Source: Standard fact; no single source recorded. (Checked)
- Black, 2 Aug 1927, Lackawanna ferry, *New York Times* page. Source: Black, *IEEE Spectrum* Dec 1977. (Checked)

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