Adding without carrying
The nim-sum of the heap sizes: write them in binary and add each column separately, discarding any carry. A column with an even number of ones cancels. The position is lost for the player to move exactly when every column cancels.
10 essays call
nim-sum. The drawing above is what it returns with no arguments at all; every
call below passes it something, because a placement that passes nothing draws whichever
member of the family the generator happens to default to rather than the one its essay argues
about.
The positions it draws
11 distinct positions, harvested by drawing the figure again at the options each essay asked for.
Where it is called
Every one of these figures is a drawing of the same construction, so a correction to it corrects them all.
Impartial games
Taking from several heaps at once
Moore's Nim lets a move take from as many as k heaps at a time, and the losing positions are still read off the binary columns — divisible by k + 1 rather than by two. The rule agrees with exhaustive search over 54,264 positions and never disagrees, and it decides every outcome while supplying no value at all: reading the same columns as a base-3 number gets the Grundy value right on 42 of 330 positions.
Impartial games
No two heaps alike
Welter's game is Nim with one extra clause — no two heaps may be the same size — and the clause is fatal to the nim-sum, which gives the right answer in none of the 120 three-coin positions. What replaces it is a function of pairs: ⟨a | b⟩ = (a ⊕ b) − 1, exact on all 55 two-coin positions, and nim-added over every pair it is exact on the whole board provided the number of coins is even.
Impartial games
The patch that generalised
Misère Nim takes a one-line patch: play the normal-play strategy until every heap holds a single counter, then invert. Moore's Nim, where a move may take from up to k heaps at once, takes exactly the same patch with exactly the same modulus — and the two rules disagree on six positions out of 923.
Impartial games
The rule a smaller move breaks
Moore's Nim lets a player take from at most k heaps, and its winning condition is the binary columns summed modulo k + 1. Cap the amount as well and the obvious repair — reduce each heap modulo the cap plus one, then read the columns — is exact at every cap when k is one and wrong at every cap when k is two or three. The reason is stronger than a broken rule: at k ≥ 2 the residues do not determine the outcome at all, so nothing of that shape can work.
Impartial games
The wider move is the easier game
An earlier essay ruled out every rule that reduces the heaps and reads the residues, and asked for a two-part statistic: the residues plus one more count. Four second parts are tested here and none of them decides. What turns up instead contradicts the premise the request was made under — a move that may reach three heaps is more predictable than one that may reach two, on every cap, every candidate rule, and after the change in the base rate is taken out.
Impartial games
The count of odd heaps
The rung below refused a family of two-part rules for bounded Moore's Nim and asked what the 364 losing positions have in common as a set. They have an invariant, and it is a statistic of the whole position rather than of a heap: how many heaps hold an odd number. Every all-even position is lost, at every width of move, by a restoring strategy — and the count settles every position at one heap a move and at four, and a little over half at two.
Impartial games
The parities, in size order
The rung below settled four of six parity classes in bounded Moore's Nim and asked whether the sizes pick out the losing positions in the two it could not. They do — but only through the order they put the parities in. Sort the heaps largest first, read off their parities, and that five-bit word settles the whole game at every width of move, with the losing words forming a subspace.
Impartial games
The parameter was the difference
The losing words of bounded Moore's Nim form a linear subspace and no map was known whose kernel they are. The equations exist, four conditions cover all thirteen cases at three to six heaps, and they are indexed not by the heap count but by the heaps less the width of a move — which turns the failure at six heaps into a prediction about seven.
Impartial games
The family with two witnesses
Six predictions about seven heaps were written down and deliberately not run. Five of them held. The one that broke is the condition that had been checked against two cases when it was proposed — the fewest of the four — and at seven heaps it does not merely give the wrong answer, it asks a question the parity word has stopped being able to answer.
How it was found
A machine that knew one theorem
In 1940 a machine of relays played Nim against the public at the New York World's Fair, on four rows of up to seven lamps, and is reported to have won about nine games in ten. Everything it knew was Bouton's theorem, which is three column parities and one test per row. The record is the interesting number, and it is a measurement of the visitors rather than of the machine: against a visitor choosing at random the machine wins 98.5 per cent, and nine in ten is what a visitor earns by playing perfectly once seven lamps or fewer are lit. On the board the game is usually set up on, a visitor handed the first move cannot win at all.
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