# Conkers

> A playable simulation of the British playground game conkers, built on exact
> pendulum mechanics, Hertzian contact and Weibull flaw statistics. Free, runs
> entirely in the browser, calls no model, stores nothing off-device.

https://conkers.skillsafe.ai/

## What it answers

The folk rule is "a bigger conker wins because it's stronger". The app shows the
claim is half right and backwards in its reasoning:

- A bigger conker is WEAKER on its own. At a fixed impact speed no Hertzian
  stress depends on size at all - {E, rho, v} contain no length - so a bigger
  conker takes the same stress over a larger stressed volume, and weakest-link
  statistics give strength proportional to V^(-1/m).
- A bigger conker is SAFER in a duel. One contact force is divided by two
  cross-sections, and the bigger body divides it by more. Risk of rupture goes as
  R^(3-2m), so the odds it outlives the other are (R1/R2)^(2m-3).
- The crossover is exactly m = 3/2. Below it the sign flips and the bigger conker
  is the one that breaks.
- The strength of the folk rule therefore rests entirely on m, the Weibull
  modulus of horse-chestnut seed, which NOBODY HAS EVER MEASURED. Biological
  brittle solids run from 1.8 (nacre) to 4.5 (dentin). At m = 1.8 a 30% size
  advantage is worth 1.17:1; at m = 4.5 it is worth 4.83:1.

## What was verified, and how

- Pendulum closed forms against RK4: arrival speed to 2e-9 relative, quarter
  period to 6e-13, with the complete elliptic integral computed by the
  arithmetic-geometric mean and cross-checked against its hypergeometric series.
- The Hertz impact-duration constant three ways that share no code: a Beta
  function (2.9432752), tanh-sinh quadrature, and RK4 on the contact ODE. The
  resulting prefactor 2.8682657 reproduces the 2.87 quoted in the literature.
- The size exponent -1/m recovered from a flaw simulation that was never told the
  Weibull law, with bootstrap 95% intervals, across four flaw populations plus a
  control.
- The documented relation m = 2n-2 (a flaw-SIZE tail exponent n gives a strength
  modulus 2n-2) recovered from a sampler told only n.
- A control that could genuinely fail: holding the flaw COUNT fixed instead of the
  flaw DENSITY removes the size effect entirely, and its interval straddles zero.
- The duel exponent 2m-3 recovered from single-strike simulations at m = 2, 3 and
  4.5, at the crossover m = 3/2 where it must vanish, and below it at m = 1.15
  where the sign must flip.

## Corrections worth knowing

- Weibull's 1951 paper does NOT state the volume-scaling law. It gives the
  weakest-link form; the explicit sigma proportional to V^(-1/m) is a later
  standard derivation. Cite Quinn & Quinn 2010 or Shin & Jang 2019 for it.
- There is no published physics of conkers. OpenAlex returns ten works titled
  "conkers" - sound art, pet toxicology, theology, a risk editorial. arXiv
  returns zero.
- No hardening method has ever been measured. Not baking, not vinegar, not a year
  in a drawer, not nail varnish. Every source asserts; none measures.
- The World Conker Championships have no rule against hardening. Rule 1 makes it
  impossible instead: every conker is supplied by the organisers and drawn blind.

## Sections

- Play - a real duel against the engine, with a strike log
- The swing - the exact pendulum, and why mass does not appear in it
- The impact - Hertzian contact, the duration constant, and why mismatched
  conkers hit each other more gently
- Does size win? - the size effect, the duel exponent, the crossover, the control
- Rules & about - the ten championship rules, the folklore, and a provenance
  table for every number on the site

## Terms

Free. No account, no model call, no credits, no tracking, no network requests
after the page loads. An independent reimplementation; not affiliated with the
World Conker Championships or the Ashton Conker Club.