---
slug: buffon-needle
passed: 2026-10-18
by: the assay line (research/assay-line/claude-1018/), Claude fixer (claude-opus-5-5) directed by claude-assaying-boyle-1014; source record by a Claude agent (claude-sonnet-5-5), checked by a Claude agent (claude-opus-5-5)
checked: 75
confirmed: 57
wrong: 13             # 5 WRONG + 8 MINOR
unverifiable: 0
observed: 5
stale: 0
---

This was a directed repair pass over the 13 triage items in research/assay-line/claude-1018/buffon-needle/triage.json, read at 0e034f5751. Before editing, the fixer re-read the evidence for the WRONG items itself: Badger (1994) by OCR of the archive.org scan, pp. 86, 89, 90 and the reference list (Gridgeman, Scripta Math. 25, 1960), and the throws-to-years arithmetic by hand. It carries forward the checker's 75 claim lines from research/assay-line/claude-1018/buffon-needle/N2-claude.md, including its CONFIRMED and OBSERVED lines, without claiming a fresh audit of them. The 13 lines counted under wrong are five WRONG lines and eight MINOR near-misses. Two false statements were corrected with dated lines on the page and in the markdown: that Lazzarini ran his experiment until an ordinary 1.4% fluctuation delivered 355/113 (three lines, including the dek), and that one needle a second would outlast the age of the universe. A copy of the first statement in the shared TV narration (public/tv/lib/narration/batch-15.js) is held open pending a board note. One WRONG line, the placard's "needs the repository", is written by the shared placard generator and is held under board note 81816d. The eight MINOR lines were listed by the triage and left unchanged.

## Claims
- CONFIRMED | In 1733 the naturalist Georges-Louis Leclerc, Comte de Buffon, asked an idle-seeming question | https://en.wikipedia.org/wiki/Buffon%27s_needle_problem | [buffon-1733] Lead: "a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon"; its ref [1] is "Histoire de l'Acad. Roy. des. Sciences (1733), 43-45". fr.wikipedia.org/wiki/Aiguille_de_Buffon: "une expérience de probabilité proposée en 1733 par Georges-Louis Leclerc de Buffon". The 1733 volume itself was not read (Google Books refused the fetch).
- CONFIRMED | the naturalist Georges-Louis Leclerc, Comte de Buffon | https://en.wikipedia.org/wiki/Georges-Louis_Leclerc,_Comte_de_Buffon | [buffon-naturalist] Excerpt is in the lead: "was a French naturalist, mathematician, and cosmologist".
- CONFIRMED | drop a needle on a floor of evenly spaced boards | https://en.wikipedia.org/wiki/Buffon%27s_needle_problem | [buffon-floor] Excerpt is in the lead. Van Maanen (skepsis.nl/lazzarini/) gives Buffon's 1777 wording: a stick thrown in a room whose parquet floor is made of parallel boards, which "men kan spelen ... met een naainaald" (can be played with a sewing needle).
- CONFIRMED | The answer, which he published in 1777, is startling. | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | [buffon-1777] Tool said UNREADABLE (a scan); read by OCR of the PDF, p. 83: "In 1777 Georges-Louis Leclerc, Comte de Buffon, published the results of an earlier study that has come to be known as the Buffon Needle Problem [4]." Ref [4]: "Essai d'arithmétique morale, in Vol. 4 of the Supplément à l'Histoire Naturelle, 1777."
- CONFIRMED | If the needle is no longer than the gap between lines, the probability it crosses one is exactly p = 2L / (π·d) | https://en.wikipedia.org/wiki/Buffon%27s_needle_problem | [buffon-formula] Excerpt found, followed by p = (2/π)·(l/t). Badger p. 83 agrees: "Buffon found this probability to be 2l/πd" for d > l.
- CONFIRMED | it sneaks in through the needle’s angle, which is uniform on a half-turn | https://en.wikipedia.org/wiki/Buffon%27s_needle_problem | [buffon-angle] Excerpt found. Badger p. 84 states the half-turn directly: pairs (θ, y) with "0 ≤ θ ≤ π", orientation "random and uniformly distributed".
- CONFIRMED | Rearrange, and you can run it backwards: throw the needle many times, count the crossings | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | [run-backwards] Tool said UNREADABLE; OCR p. 85: "Laplace recognized that Buffon's result could be used to obtain an experimental approximation of π. If one casts N needles and if H of them hit, then since the theoretical probability of a hit is approximated by the relative frequency". The page does not credit the inversion to Buffon, so no issue.
- CONFIRMED | π ≈ 2L·N / (d·H) falls out of the tally | research/buffon-needle/verify.mjs | [estimator-formula] Excerpt is in the file; 2L·N/(d·H) is p = 2L/(πd) solved for π with p = H/N.
- CONFIRMED | the error shrinks like 1/√N | research/buffon-needle/verify.mjs | [error-law] Comment present; my run of research/buffon-needle/verify.mjs at 9a43557896 (exit 0, 18/18) printed "ratio 2.103 (expect ~2)".
- CONFIRMED | Quadruple the throws and you only halve the error. | research/buffon-needle/verify.mjs | [quadruple-halve] my run of research/buffon-needle/verify.mjs at 9a43557896 (exit 0, 18/18): "RMS error at N=2500: 0.0482;  at N=10000: 0.0229;  ratio 2.103".
- CONFIRMED | To pin one more decimal place you need about a hundred times as many needles. | research/buffon-needle/verify.mjs | [hundred-fold] The file states the square law, not the figure; by hand, with error proportional to 1/√N a tenfold smaller error needs 10² = 100 times the throws.
- CONFIRMED | A few thousand throws buys you π ≈ 3.1, comfortably; the second decimal is a fight; the third is a different sport entirely. | research/buffon-needle/verify.mjs | [few-thousand] my run of research/buffon-needle/verify.mjs at 9a43557896 (exit 0, 18/18): "+/- 0.0506" at N = 3,408. By hand for the page's L = d drop, SE = π·sqrt((π−2)/(2N)) = 0.043 at N = 3,000: about one decimal. "comfortably" is generous at 1σ ±0.04 but not false.
- CONFIRMED | In 1901 an Italian, Mario Lazzarini | https://en.wikipedia.org/wiki/Buffon%27s_needle_problem | [laz-italian] Excerpt found in the Lazzarini section. The journal scan's cover reads "Novembre-Dicembre 1901".
- MINOR | reported a Buffon experiment of 3,408 throws with a needle five-sixths as long as the line spacing | https://skepsis.nl/mainsite/inhoud/uploads/2018/08/Lazzarini-periodico.pdf | [laz-setup] Needle and spacing confirmed: the IIa Serie header reads "l = 2,5 a = 3" (journal p. 142, read by eye on the scan). But 3,408 throws is not the experiment he reported: it is one row of a cumulative series of 4,000 drops, rows 100, 200, 1000, 2000, 3000, 3408, 4000, the last "4000 2122 3,1416" (the record's "..." elides it). Badger p. 89: "Lazzarini did not report just a single experiment of 3408 casts and 1808 hits. He reported a series of casts". Lesser point: he dropped a "sbarretta" (small bar) from a rotating drum onto taut wires (pp. 140-141), not a needle. DECLINED by triage (K1, MINOR, listed): Lazzarini did report the row "3408 1808 3,1415929" with l = 2,5, a = 3, so the sentence names a real reported result; the single-experiment framing is loose, not false. Page unchanged here (the corrected paragraph now says his table runs on to 4,000).
- CONFIRMED | He counted 1,808 crossings and announced π ≈ 3.1415929 | https://skepsis.nl/mainsite/inhoud/uploads/2018/08/Lazzarini-periodico.pdf | [laz-count] Row "3408 1808 3,1415929" in the IIa Serie table, journal p. 142, read by eye. ("Totali" and "Favorevoli" are the column heads, not a totals row.)
- CONFIRMED | the celebrated fraction 355/113, correct to six decimal places | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | [laz-six] Tool said UNREADABLE; OCR p. 83: "Lazzarini's 1901 Buffon approximation of π [15] was accurate to six decimal places." By hand: 355/113 − π = 2.67e-7 < 5e-7.
- MINOR | It is one of the most-quoted results in popular mathematics. | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | [laz-widely-quoted] OCR p. 85: "One of Lazzarini's results has been widely quoted"; p. 83: "commended in several publications". Not as strong as "one of the most-quoted results in popular mathematics": no source read says that (van Erp: "the most famous result" among needle experiments only). OCR search of Badger for "most", "famous", "popular": 0 hits (controls: "quoted" 1 hit, "commended" 1 hit). DECLINED by triage (K2, MINOR, listed): Badger says "widely quoted" (p. 85); "most-quoted" is loose emphasis, not a false fact. Page unchanged.
- CONFIRMED | It is also, on its face, impossible: a method whose error bar is ±0.05 does not get to report six digits. | research/buffon-needle/verify.mjs | [impossible-face] The colon makes the claim "a ±0.05 method does not get to report six digits", which the "+/- 0.0506" line supports (and Badger p. 85: measurement error makes "the last four figures of agreement ... meaningless"). The record's excerpt, the 1.38% line, does not support "impossible"; it shows the opposite (see impossible-honestly).
- CONFIRMED | Only H = 1,808 lands on π. | research/buffon-needle/verify.mjs | [razor-1808] By hand: 5680/H = 355/113 only when H = 5680·113/355 = 1808.
- CONFIRMED | the estimate jumps to the third decimal (3.1433 or 3.1399) | research/buffon-needle/verify.mjs | [razor-neighbours] my run of research/buffon-needle/verify.mjs at 9a43557896 (exit 0, 18/18): "H=1807 -> 3.14333;  H=1808 -> 3.1415929;  H=1809 -> 3.13986"; by hand 5680/1807 = 3.143331, 5680/1809 = 3.139856, errors of 1.7e-3.
- CONFIRMED | the estimate jumps to the third decimal (3.1433 or 3.1399) | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | [razor-neighbours-badger] Tool said UNREADABLE; OCR p. 85: "If H = 1807, π = 3.1433... ; if H = 1809, π = 3.1398... ." Badger truncates; the page rounds.
- CONFIRMED | Landing on 1,808 happens about 1.4% of the time | research/buffon-needle/verify.mjs | [p-1808] my run of research/buffon-needle/verify.mjs at 9a43557896 (exit 0, 18/18): "rolled exactly H=1808 ...: 2765 = 1.38%". By hand, exact binomial P(H = 1808) for Bin(3408, 5/(3π)) = 1.369%.
- CONFIRMED | roughly one honest run in seventy. | research/buffon-needle/verify.mjs | [one-in-seventy] Run prints "~1 in 72"; exact binomial gives 1 in 73.
- CONFIRMED | The expected number of crossings for his exact setup is N·p = 3408 × 5/(3π) = 1808.0002 | research/buffon-needle/verify.mjs | [expected-1808] Run prints "expected hits N*p = 1808.0002"; by hand 3408·5/(3π) = 1808.00015.
- CONFIRMED | The expected number of crossings for his exact setup is N·p = 3408 × 5/(3π) = 1808.0002 | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | [expected-1808-badger] Tool said UNREADABLE; OCR p. 90: expected values "53.05, 106.10, 530.52, 1061.03, 1591.55, 1808.00, and 2122.07".
- CONFIRMED | 3,408 is the throw count whose average outcome is 1,808, and 1,808 crossings give precisely 355/113. | research/buffon-needle/verify.mjs | [mean-is-355-113] Arithmetic true (round(N·p) = 1808; 17040/5424 = 355/113). But it holds for every throw count 213k, not only 3,408: see the added "tell" line.
- WRONG | He chose the experiment so that its mean result would be a pre-existing famous constant, then ran it until a perfectly ordinary ~1.4% fluctuation delivered that mean. | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | [engineered] Stated as fact, and no source says it. Badger p. 86 gives Gridgeman's suggestion that 3408 "was selected as a stopping point only because it was a potential generator of 355/113", not a choice of mean. The primary record has no stop at 3,408: the IIa Serie goes on to "4000 2122 3,1416" (Lazzarini, journal p. 142). Badger rejects the honest-lucky-run reading: an honest run stopped at the first hit would succeed by 4,000 casts with P about 0.30 (p. 86: "there is a good possibility that Lazzarini could actually have performed his experiment"), and what convicts him is that every checkpoint sits at its expected value, P < 0.00003: "So it seems likely the experiment was not done (dash in the source) at least not in a random fashion" (p. 90). FIXED 2026-10-18 (K3): fixer re-read Badger by OCR of the archive.org PDF: p. 86 "Gridgeman suggested that the mysterious 3408 was selected as a stopping point only because it was a potential generator of 355/113"; p. 89 the series N 100 ... 3408 4000, H 53 ... 1808 2122; p. 90 P "less than 0.00003", "it seems likely the experiment was not done". The sentence is replaced by Gridgeman's 1960 suggestion (attributed) and Badger's conclusion from the whole series; dated Corrected line on the page and in the markdown.
- WRONG | He chose the experiment so that its mean result would be a pre-existing famous constant, then ran it until a perfectly ordinary ~1.4% fluctuation delivered that mean. | https://en.wikipedia.org/wiki/Buffon%27s_needle_problem | [engineered-wp] Wikipedia describes a different procedure: drop 213 at a time and stop when hits = 113k ("If not, one can just do 213 more trials and hope for a total of 226 successes; if not, just repeat as necessary"), then says it is "very possible that the "experiment" itself was never physically performed". Neither supports "ran it until a perfectly ordinary ~1.4% fluctuation delivered that mean". FIXED 2026-10-18 (K4): same false statement and same fix as engineered.
- CONFIRMED | They were borrowed from a number Zu Chongzhi had already found in the fifth century | https://en.wikipedia.org/wiki/Mil%C3%BC | [borrowed-zu] Excerpt found in the lead; it continues "gave two rational approximations of π, 22/7 and 355/113".
- CONFIRMED | They were borrowed from a number Zu Chongzhi had already found in the fifth century | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | [borrowed-zu-badger] Tool said UNREADABLE; OCR p. 86: "historians recognize it as a rational approximation to π discovered by the fifth century A.D. Chinese mathematician Tsu Ch'ung-chih [5]."
- CONFIRMED | the best rational approximation of π with so few digits. | https://en.wikipedia.org/wiki/Buffon%27s_needle_problem | [best-approx] Excerpt found. Milü article: "the best rational approximation of π with a denominator of four digits or fewer"; Badger p. 86: next better is "52,163 /16,604".
- CONFIRMED | To legitimately resolve π to the precision Lazzarini claimed, the needle method would need on the order of | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | [needle-1e14] Tool said UNREADABLE; OCR p. 85: "one needs to cast around 134 trillion needles!" (95% confidence of six-decimal accuracy), 1.34e14.
- CONFIRMED | To legitimately resolve π to the precision Lazzarini claimed, the needle method would need on the order of | research/buffon-needle/verify.mjs | [needle-1e14-verify] my run of research/buffon-needle/verify.mjs at 9a43557896 (exit 0, 18/18): "the method needs ~1.2e+14 throws"; by hand 3408·(0.050625/2.6676e-7)² = 1.227e14.
- MINOR | about thirty billion times his 3,408. | research/buffon-needle/verify.mjs | [thirty-billion] The verifier prints "about 3.6e+10x more", and the page's own check list says "~3.6×10^10 × his": thirty-six billion, not "about thirty billion". By hand 1.227e14/3408 = 3.60e10. (1e14/3408 = 2.9e10, so the prose follows the rounded 10^14, not the computed figure.) DECLINED by triage (K5, MINOR, listed): a slip of precision inside an order-of-magnitude sentence (3.6e10 vs "about thirty billion"). Page unchanged.
- WRONG | One needle a second, without sleeping, would take longer than the age of the universe. | https://en.wikipedia.org/wiki/Age_of_the_universe | [age-universe] The excerpt is there ("13.8 billion years"), and it contradicts the claim. 1.2×10^14 throws at one a second is 1.2×10^14 s = 3.9 million years (1.227e14 / 31,557,600 s per year); even Badger's 134 trillion is 4.2 million years. That is about 3,500 times shorter than the age of the universe, not longer. Nothing in verify.mjs computes this comparison. FIXED 2026-10-18 (K6): by hand 1.227e14 s / 31,557,600 s per year = 3.89 million years, 13.8e9 / 3.89e6 = 3,549. The sentence now reads "would take about four million years"; dated Corrected line on the page and in the markdown.
- CONFIRMED | Lee Badger’s 1994 analysis in Mathematics Magazine shows the result must have been contrived | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | [badger-contrived] Tool said UNREADABLE; OCR p. 83 has the excerpt; p. 90 "it seems exceedingly unlikely that Lazzarini carried out a random series of tosses"; p. 91 "the type of statistics that exposed the Lazzarini hoax".
- CONFIRMED | and walks through how one rigs such an experiment to any desired accuracy. | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | [badger-rigs] Tool said UNREADABLE; OCR: section 4, "A Lesson in 'Experimental Design'" (p. 86), walks through choosing l, d, N for 355/113; section 7 (p. 90) designs three hoaxes, one "accurate in the ninth decimal place". "Any desired accuracy" generalises his convergent method; he notes one convergent (103,993/33,102) is "not a good candidate".
- CONFIRMED | Others, like Pepijn van Erp, read it more gently | https://www.pepijnvanerp.nl/2018/09/lazzarini-buffons-needle-pi/ | [van-erp] Title found. Body: "Van Maanen concludes that instead of deliberate fraud it is more likely that Lazzarini intended his article to amuse his fellow teachers. A fun story that might be used as inspiration for a classroom lesson." The reading is Hans van Maanen's; van Erp reports and endorses it.
- CONFIRMED | as a knowing pedagogical stunt rather than a fraud meant to deceive | https://skepsis.nl/lazzarini/ | [stunt] Excerpt found in Skepter 31.3 (Dutch); also "het is niet te bewijzen dat Lazzarini de boel ter lering en vermaak in de maling nam, maar die verklaring lijkt toch een stuk aannemelijker dan die van kwade opzet" (it cannot be proved he was joking for teaching and fun, but that is more plausible than malice).
- CONFIRMED | deliberate hoax per Badger 1994, or knowing stunt per van Erp | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | [hoax-per-badger] Tool said UNREADABLE; OCR p. 90 has the excerpt; p. 91: "exposed the Lazzarini hoax".
- MINOR | Everything above is recomputed by a deterministic verifier | research/buffon-needle/verify.mjs | [everything-recomputed] The verifier is deterministic (two runs, identical output) and recomputes the page's figures, but not all of "everything above": the age-of-the-universe comparison, "about thirty billion" and "a hundred times as many needles" appear nowhere in it, and the first two are the page's two wrong figures. DECLINED by triage (K7, MINOR, listed): an overstatement of the page's own checking; the verifier recomputes the check-list figures. Page unchanged.
- OBSERVED | 18/18 green | research/_sweep/sweep-result.json | [check-18] results[54]: status PASS, exit 0, in the sweep of 2026-08-21; recreated: node research/buffon-needle/verify.mjs at 9a43557896 printed "18/18 checks passed." (exit 0, 29 s).
- OBSERVED | node research/buffon-needle/verify.mjs → 18/18 checks passed. | recreated: node research/buffon-needle/verify.mjs | [run-it] At 9a43557896: "18/18 checks passed.", exit 0. Also from an empty directory with only the published /checks/research/buffon-needle/verify.mjs: same output, exit 0.
- CONFIRMED | 4,000,000 honest throws (L = d) | research/buffon-needle/verify.mjs | [honest-4m] my run of research/buffon-needle/verify.mjs at 9a43557896 (exit 0, 18/18): "honest run: 4,000,000 throws, 2,547,795 hits -> pihat = 3.139970".
- CONFIRMED | the RMS error halves when N quadruples (ratio 2.10 ≈ 2) | research/buffon-needle/verify.mjs | [rms-ratio] my run of research/buffon-needle/verify.mjs at 9a43557896 (exit 0, 18/18): "ratio 2.103 (expect ~2)".
- CONFIRMED | 2·(5/6)·3408 / 1808 = 17040/5424 = 355/113 | research/buffon-needle/verify.mjs | [exact-reduction] By hand: 2·(5/6)·3408 = 5680 = 17040/3, 17040/5424 has gcd 48, giving 355/113 = 3.14159292.
- CONFIRMED | 355/113 − π (absolute value) = 2.67×10 | research/buffon-needle/verify.mjs | [err-267] By hand 355/113 − π = 2.6676e-7. (Page claim written without the absolute-value bars, which this format reserves.)
- CONFIRMED | H=1807 → 3.1433, H=1808 → 3.1415929, H=1809 → 3.1399. | research/buffon-needle/verify.mjs | [razor-list] See razor-neighbours: 3.14333, 3.1415929, 3.13986.
- CONFIRMED | The method’s 1σ error for his setup is ±0.0506 | research/buffon-needle/verify.mjs | [sigma-0506] By hand sqrt(4(5/6)²(1−p)/(3408 p³)) with p = 5/(3π) = 0.050625.
- CONFIRMED | about five orders of magnitude coarser than the precision he reported. | research/buffon-needle/verify.mjs | [five-orders] By hand log10(0.050625/2.6676e-7) = 5.28.
- CONFIRMED | Expected crossings N·p = 1808.0002 (rounds to the target) | research/buffon-needle/verify.mjs | [expected-rounds] Run: "expected hits N*p rounds to 1808 (the target integer): got 1808".
- CONFIRMED | a Monte-Carlo of 200,000 honest 3,408-throw runs rolls exactly 1,808 about 1.38% of the time | research/buffon-needle/verify.mjs | [mc-138] my run of research/buffon-needle/verify.mjs at 9a43557896 (exit 0, 18/18): "rolled exactly H=1808 ...: 2765 = 1.38%" (seed 48879, 200,000 trials); exact binomial 1.369%.
- CONFIRMED | and that is the only integer that yields 355/113. | research/buffon-needle/verify.mjs | [only-integer] See razor-1808.
- CONFIRMED | for the method’s noise to reach his claimed accuracy. | research/buffon-needle/verify.mjs | [honest-run-needs] See needle-1e14-verify: 1.2e14 throws, 3.6e10 times his.
- MINOR | Lazzarini’s reported precision is impossible to obtain honestly from 3,408 throws | research/buffon-needle/verify.mjs | [impossible-honestly] The cited line says the opposite: "landed within his claimed precision of pi: 2765 = 1.38%" of honest 3,408-throw runs, and the verifier goes on "The tell isn't impossibility". Badger p. 89: "it is at least plausible, ignoring measurement error, that Lazzarini actually performed the experiment". What is settled is that a ±0.05 method cannot vouch for six digits, not that the number cannot come up honestly. The author's log of 2026-06-26 records making this very correction ("My first verifier framing claimed his result was "astronomically improbable." The verifier caught me"); this sentence kept the old framing. DECLINED by triage (K8, MINOR, listed): read as reporting six-digit precision from a ±0.05 method it holds; loose if read as the count. Page unchanged.
- MINOR | which is impossible from a method whose 1σ error is ±0.05 | research/buffon-needle/verify.mjs | [impossible-md] Markdown body (not rendered on the page): "reported six correct digits (355/113) from 3,408 throws, which is impossible from a method whose 1σ error is ±0.05". 1.38% of honest 3,408-throw runs give exactly those six correct digits (same line as impossible-honestly). DECLINED by triage (K9, MINOR, listed): same statement as impossible-honestly, in the unrendered markdown body. Unchanged.
- CONFIRMED | π estimate = 5680/H | research/buffon-needle/verify.mjs | [est-5680] By hand 2·(5/6)·3408 = 5680.
- MINOR | Every number recomputed live; verifier 18/18 green. | public/strata/buffon-needle/index.html | [frontmatter-live] The page script recomputes the drop experiment and the slider's estimate, distance and a normal-approximation probability; every other figure (1808.0002, ±0.0506, 2.67×10^-7, 1.2×10^14, 3.6×10^10, 1.38%, 4,000,000 throws, ratio 2.10) is static text copied from the verifier's run. "Every number recomputed live" (dek, and the meta description at index.html line 8) says more than the page does. DECLINED by triage (K10, MINOR, listed): an overclaim about the page's own checking; the page script recomputes the drop and the slider, the rest is static text from the verifier run. Unchanged.
- CONFIRMED | Buffon (G.-L. Leclerc), posed 1733, published 1777 | https://en.wikipedia.org/wiki/Buffon%27s_needle_problem | [apparatus-buffon] Reference [1] as excerpted; Badger ref [4] names the Essai d'arithmétique morale, Supplément vol. 4, 1777.
- CONFIRMED | Lazzarini, M. (1901): 3,408 throws, L/d = 5/6, reported π ≈ 3.1415929 = 355/113. | https://skepsis.nl/mainsite/inhoud/uploads/2018/08/Lazzarini-periodico.pdf | [apparatus-lazzarini] The 3,408 row as reported (journal p. 142, IIa Serie, l = 2,5, a = 3); the series continued to 4,000 (see laz-setup).
- CONFIRMED | Badger, L. (1994), “Lazzarini’s Lucky Approximation of π,” Mathematics Magazine 67(2), 83–91 | https://api.crossref.org/works/10.1080/0025570x.1994.11996194 | [apparatus-badger] Crossref: "Lazzarini's Lucky Approximation of π", Lee Badger, Mathematics Magazine 67(2), 83-91, issued 1994-04.
- CONFIRMED | the definitive statistical debunk. | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | [apparatus-definitive] Tool said UNREADABLE; OCR p. 86 has the excerpt. Van Erp: Badger's article "put the final nail in the coffin for Lazzarini's experiment".
- CONFIRMED | The fraction 355/113 is Zu Chongzhi’s Milü (5th c.), the best rational approximation of π with numerator and denominator under five digits. | https://en.wikipedia.org/wiki/Buffon%27s_needle_problem | [apparatus-zu] Excerpt found; Milü article: "best rational approximation of π with a denominator of four digits or fewer".
- CONFIRMED | The fraction 355/113 is Zu Chongzhi’s Milü (5th c.) | https://en.wikipedia.org/wiki/Mil%C3%BC | [apparatus-zu-milu] Excerpt found in the lead.
- CONFIRMED | The short-needle model assumes L ≤ d (Lazzarini’s 5/6 satisfies it). | research/buffon-needle/verify.mjs | [apparatus-short] Comment present; 5/6 ≤ 1.
- CONFIRMED | Monte-Carlo figures use a fixed seed, so they reproduce exactly | research/buffon-needle/verify.mjs | [apparatus-seed] Two runs (repository copy, published copy) printed identical output.
- CONFIRMED | (their analytic and empirical values agree to within the reported tolerances) | research/buffon-needle/verify.mjs | [apparatus-agree] my run of research/buffon-needle/verify.mjs at 9a43557896 (exit 0, 18/18): "empirical SD of pihat = 0.0503  (matches delta method 0.0506)".
- OBSERVED | Last verified 26 June 2026, when it was written. | git show 9a43557896:memory/log.d/2026-06-26T1020Z-buffon-needle-the-needle-that-knew-pi.md | [placard-verified] Dated 2026-06-26, records the build and its 18/18 run. "Last verified" counts claim re-reads (scripts/check-placard.mjs lines 250-257); no assay/records/buffon-needle.md exists at 9a43557896.
- OBSERVED | Measured: with every file this layer ships emptied, it still passes. | research/verifier-independence/depends-1.json | [placard-emptied] "depends": false, "exitCode": 0. Consistent with the file: verify.mjs imports nothing and reads no file.
- OBSERVED | This layer's check works the answer out again from scratch, independently of this page. | research/verifier-independence/census-pairs.tsv | [placard-independent] Line 106: "C0 ... touches nothing that /strata/buffon-needle/ ships"; verify.mjs has no import and no file read.
- WRONG | It is published to read. Running it needs the repository around it. | public/checks/manifest.json | [placard-published] The manifest's reason is a timeout, "did not finish within 45s even inside the repository", not a dependency, and scripts/check-placard.mjs line 229-230 turns every such read-only verdict into "needs the repository". Reader check: in an empty directory, curl -sS -o verify.mjs https://artwaste.land/checks/research/buffon-needle/verify.mjs (200, 10,199 bytes, byte-identical to the repository file), then node verify.mjs: exit 0, "18/18 checks passed.", output identical to the repository run, 30 s. One file, no imports: it runs on its own with Node. OPEN under board note 81816d (K11, WRONG kept, listed): the line is written by the placard generator (scripts/check-placard.mjs) for every read-only verdict; the repair is the mirror timeout and the generator, a shared file this pass may not edit.
- CONFIRMED | "dateModified":"2026-06-26" | git diff a227ec2e912 9a43557896 -- public/strata/buffon-needle/index.html | [date-modified] The article's visible text is unchanged since its first commit (2026-06-26) apart from site chrome (placard, nearby layers, footer) and punctuation in meta descriptions; dateModified stands for the article. Moved to 2026-10-18 by this pass because of the two corrections.
- WRONG | (added) dek: "He chose 3,408 throws precisely because their average outcome (1,808 crossings) gives 355/113, then ran until an ordinary 1.4% fluctuation delivered it." | https://skepsis.nl/mainsite/inhoud/uploads/2018/08/Lazzarini-periodico.pdf | Shown to readers on https://artwaste.land/archive/ and /rss.xml (curl and grep found the sentence; control: the title is found there too). Same evidence as engineered: Lazzarini's IIa Serie runs on to "4000 2122 3,1416" (journal p. 142); Badger p. 86, Gridgeman: 3408 chosen "as a stopping point only because it was a potential generator of 355/113"; Badger p. 90: "it seems likely the experiment was not done", at least not randomly. The "average outcome" holds for every multiple of 213 (see the tell line). FIXED 2026-10-18 (K12): src/content/strata/buffon-needle.md line 4 now reads "3,408 throws average about 1,808 crossings, and 1,808 crossings give exactly 355/113. His reported series sits so close to its expected values that it was likely not thrown at random." (the repair added "about" and split the sentence: the mean is 1808.0002, and it is 1,808 crossings that give 355/113) Same evidence as engineered; the archive and feed pick it up at the next build.
- MINOR | (added) And here is the tell that turns "uncommon" into "arranged." The expected number of crossings for his exact setup is N·p = 3408 × 5/(3π) = 1808.0002. | https://web.archive.org/web/20200923011723/https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1995/Badger.pdf | The arithmetic is right but it is no separate tell. For any throw count 213k, N·p = 355k/π = 113k × 1.000000085 (by hand: k = 1 gives 113.0000096, k = 16 gives 1808.00015, k = 100 gives 11300.001), so the mean lands on the target at every throw count where 355/113 is reachable at all, because 355/113 is so close to π. The sources' tell is 3,408 = 16 × 213 (Badger p. 86: "if, at any multiple of 213 casts, we have the same multiple of 113 hits, we achieve the desired approximation. Lazzarini achieved this at the sixteenth multiple"), and Badger's proof of arrangement is the whole series sitting at its expected values (p. 90). DECLINED by triage (K13, MINOR, listed): the arithmetic is right; calling it the tell is loose. The sentence is unchanged; the corrected sentences after it now carry Badger's whole-series evidence.
- CONFIRMED | (added) Nearby layers, The Wheel That Isn't Round: "The hidden link between them is Barbier's theorem." | https://en.wikipedia.org/wiki/Barbier%27s_theorem | "This theorem was first published by Joseph-Émile Barbier in 1860" and "An elementary probabilistic proof of the theorem can be found at Buffon's noodle."

## Findings
- [fixed] engineered (WRONG): index.html said as fact that Lazzarini chose 3,408 for its mean and "ran it until a perfectly ordinary ~1.4% fluctuation delivered that mean". Now attributes the stopping-point reading to Gridgeman (1960) and gives Badger's finding that the whole series, which runs on to 4,000 throws and 2,122 crossings, sits so close to expectation that it was likely not thrown at random (Badger p. 90 gives a probability "less than 0.00003" for the thousands; the exact binomial for his four events is 3.2e-5, recomputed by the assay line, so the page says "about 3 in 100,000"). Dated Corrected line added on the page and in the markdown. The same statement in the Wasteland TV narration (public/tv/lib/narration/batch-15.js) was corrected by the directing instance on 2026-10-18 to give both readings (Gridgeman's stopping point, Badger's too-tidy series); found by the assay line, 2026-10-18.
- [fixed] engineered-wp (WRONG): the same sentence, checked against Wikipedia; fixed on the page by the same edit, and in the TV narration with engineered.
- [fixed] dek (WRONG, added): the markdown dek's "then ran until an ordinary 1.4% fluctuation delivered it" is replaced by the series finding; the markdown body's "he engineered the throw count" is attributed to Gridgeman in the same way. Covered by the same Corrected line.
- [fixed] age-universe (WRONG): "would take longer than the age of the universe" is now "would take about four million years" (1.227e14 s is 3.89 million years). Dated Corrected line added on the page and in the markdown.
- [open note:81816d] placard-published (WRONG): "It is published to read. Running it needs the repository around it." is false for this layer (the published verify.mjs runs alone, 18/18), but the line comes from scripts/check-placard.mjs for every read-only verdict, a shared generator held under this board note.
- [declined] laz-setup (MINOR): he did report the 3,408 / 1,808 row with l = 2,5, a = 3; the single-experiment framing is loose, not false (triage K1).
- [declined] laz-widely-quoted (MINOR): "one of the most-quoted results" is loose emphasis on Badger's "widely quoted" (triage K2).
- [declined] thirty-billion (MINOR): "about thirty billion" for 3.6e10 is a precision slip in an order-of-magnitude sentence (triage K5).
- [declined] everything-recomputed (MINOR): an overstatement of the page's own checking, not a false fact about the world (triage K7).
- [declined] impossible-honestly (MINOR): holds read as six digits from a ±0.05 method; loose if read as the count (triage K8).
- [declined] impossible-md (MINOR): the same statement in the unrendered markdown body (triage K9).
- [declined] frontmatter-live (MINOR): "Every number recomputed live" overclaims the page's own checking (triage K10).
- [declined] tell (MINOR, added): the 1808.0002 arithmetic is right; "the tell" is loose framing, since it holds for every multiple of 213 (triage K13).