page thirty-nine

The Pyx

Thirty-eight rooms have counted the dead, dated the timbers and named the salts, and not one of them has ever been paid. Today's number founds a mint, strikes a year of coin, and sends the sample box up to a jury who want to know which of it is false, and how few weighings it takes to be sure.

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The weighings

Put it to the jury

Say it as soon as you are sure, and not before. A guess that happens to be right is still a guess, and the jury will know.

Mint
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The arrow keys walk to other days. Every day is a different mint, a different coin and a different handful of false ones.

What is real here and what is not

The trial and the puzzles are real. The mint, its year, its officers and its false coins are the day's number.

  1. 01

    The box

    A pyx is a box. Since at least 1282 the English Mint has set aside a few coins from every batch it strikes and locked them in one, and once a year a jury of goldsmiths opens it and tests the coins against trial plates of known fineness, kept apart from the Mint for exactly that reason. Isaac Newton sat through it as Master of the Mint in 1710 and was accused of making the gold too poor; he blamed the new plate. The trial is still held at Goldsmiths' Hall every year.

  2. 02

    Three ways to fall

    A beam balance with nothing on it but coins has three answers: left, right, level. So n weighings can tell apart at most 3n possibilities. Nine coins with one known to be light are nine possibilities, so two weighings; thirteen are thirteen, so three. A false coin of unknown sign doubles them, and that bound is what makes every trial here a real question rather than a lucky one.

  3. 03

    Twelve, and then thirteen

    The twelve-coin problem went round in the 1940s; one early printed version is E. D. Schell's, in 1945. Freeman Dyson settled it in 1946: with n weighings and no true coin to hand you can find the false one and its sign among (3n − 3)/2 coins, which is twelve for three weighings, and among (3n − 1)/2 if one coin is known to be true, which is thirteen. The plate in trial III is that one coin.

  4. 04

    The honest liar

    Here the false coin is not fixed in advance. The balance always answers truthfully, but among all the coins it could be, the page lets it be whichever leaves you furthest from an answer, so no one makes par by luck. A solver that counts the coins by what they could still be finds par for each trial and marks each of your weighings. The sacks are the old steelyard trick: take a different number from each, or powers of two when any number of sacks may be light, and one reading says it all. Real light coins vary; these are all short by exactly the same weight.

Thirty-nine rooms, and one number doing all the work. Today it strikes a year of coin, slips the false ones in and waits to see whether you can weigh as if the worst were always true.