The thread: One clause decides it — page 3
Impartial games
What the numerals knew
Every factor in the Fibonacci Nim family gives a numeral system, and the rung below predicted its separation condition would be the lag of the recurrence the losing heaps satisfy. It is not. The gap is the factor — one at c = 1, Zeckendorf's two at c = 2, and c at every factor to eight — while the lag goes 1, 2, 4, 6, 8, 11, 14, 17 and leaves its own pattern at six. The numerals then solve every one of 194,480 states, cap and all.
Impartial games
A code that climbs by three
Five hexadecimal codes were known to repeat with a constant added, and every one of the five constants was a power of two — either a fact about exclusive-or or a coincidence over five cases. Sweeping all 255 two-digit codes settles it: seventy-one climb, seventy of them by 1, 2, 4 or 16, and one by three. The exception is ·3f, whose values are 3⌊n/6⌋ + (n mod 3) on every heap to twelve hundred.
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.
Particular games
The condition that survived the wider sweep
Which pairs of subtraction lists have a value sequence that repeats? Over the 49 pairs drawn from one, two and three, two conditions answer it identically — a translation and all-odd — and both are exactly right. Over the 961 pairs drawn from one to five, all 83 translations still repeat with no exception and four all-odd pairs do not, at heap ninety with a period as long as forty-two. Neither condition is necessary: 104 pairs repeat that satisfy neither.
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 third digit
The rung below found 71 of the 255 two-digit hexadecimal codes repeating with a constant added rather than exactly, and asked whether the same share holds one digit wider. It rises. Of the 4,095 three-digit codes, 1,433 climb and 617 repeat exactly — seven in ten of the settled ones — so a saltus is the ordinary way a hexadecimal game settles and the exact repetition the octal survey was built to find is the special case.
Particular games
The size of a cake
Ω gives the sign of a Maundy Cake and says nothing about the size, and the rung below left four values — 7, 10, 13 and 16 — unaccounted for. For a one-row cake they are a formula: write the prime factors largest first and add up their running products. The rule behind it is greedy — cut by the largest prime — and it is exact on every one-row cake to two hundred and wrong on a fifth of the two-sided ones.
Impartial games
A check in front of a search
The rung below found a pairing one move away on 288 of the 767 even first-player shapes, and asked what a solver that tested for one before recursing would save on a real game. On an even Cram board it saves nearly the whole search — a 4 × 5 board takes 17,348 node expansions without the check and one with it — and the depth profile shows why that number flatters: the check settles every winning position at the opening and at the last two moves, and about one in ten in between.
Particular games
A numeral in the empty squares
The rung below ruled out a quantitative criterion for Push and asked for a numeral over the coins combined with a count over the gaps. The two ingredients are the right way round: the colours pick a fraction — −1, −1/3, −1/7, −1/15 — and the empty squares give the binary precision, so a run of k coins before one of the other colour with g gaps is worth exactly (1 − 2^(−kg)) ÷ (2^k − 1). And it does not compose: a strip of two runs is not the sum of them, on any pair tried.
Particular games
A region one player owns
On a strip, a region containing only one player's amazons is worth exactly its free-square count, on all 45,057 positions of the rung below's sweep — and it predicted the exactness would fail in two dimensions, where an amazon can be short of room in one direction and not another. It does not fail. Over 2,412 two-dimensional regions there is no exception, and the reason is one clause: an amazon may shoot back at the square it has just left.
Particular games
The cliff a cut invents
The rung below asked for a correction term in the gap between two Push runs. There is none, because the gap's contribution vanishes: widen it and the strip's value converges geometrically, at a rate set by the back run's length alone, to a limit that is not the sum. And Shove — whose reading is exact everywhere — fails at the same cut, which says the broken thing is the cut and not the game.
Particular games
Cut small unless you are behind
The rung below found the greedy rule — cut at the largest prime — wrong on 104 of 552 Maundy Cakes and asked for a description of them. On all 104 the best cut is at the smallest prime, the exact opposite. A middle divisor is never needed on any cake in a sixty by sixty grid, and which of the two extremes wins is decided by Ω alone: cut small when Ω(m) + 1 ≥ Ω(n), large otherwise, and that is exact on all 3,540.
Impartial games
The only way to split into three
Nineteen three-digit hexadecimal codes climb by three, and the rung below asked whether they share a form and what digits they have in common. The digits are exact: on eighteen of them the only way to split a heap into three is by taking exactly three counters, and taking three counters can do nothing else. The form is not shared — the eighteen carry four distinct sequences, and exactly one of the four counts in base three.
Particular games
The short side only says how many
The rung below settled which cut to make in a Maundy Cake and left the value open. With the cut settled the recursion is a walk, the walk unrolls, and what it unrolls into is the running products of the long side's prime factors, largest first. The short side never enters the products at all — it decides how many of them there are and nothing else, so sixty-two different short sides give one value.
Impartial games
A pairing, and the pairing
The rung below repaired the half-turn check and asked whether a reflection would fire where it does not. It does — forty positions of 58,830 on the largest board — and it is sound, and it is worth one node in a thousand to a solver. It can never fire on an empty rectangle at all, which is why the ladder's whole subject is the half turn.
Particular games
The short side is not in the lemma
The closed form for a two-sided Maundy Cake rested on one unproved statement: that no divisor beats the largest prime. Written out, that statement never mentions the short side — it is an inequality between a multiset of primes and a term count — and once it is stated that way it has a two-line proof, term by term. The ladder ends in a theorem rather than a grid.
Particular games
Two strips that end the same way
If a Push strip's sensitivity is governed by its last run, then two strips agreeing at the far end should behave the same however different their fronts. On 78 of 80 tails they do, exactly. On two of them a single empty square in the prefix reaches across a gap that grows without bound and halves the rate — and the run reading turns out to be sound in one direction only.
Where it stops
The clause that turns the class off
Three rungs failed to find the dead-ending class doing measurable work, and each time the population was blamed. Toads and Frogs with and without the jump is the matched pair the anchor wanted — the same board with the class switched on and off — and on it the test the class licenses gains less from the class than a control that has never heard of it.
Impartial games
The dual was the value table
A coin-turning game's losing rows form a linear code, and a code has a dual that nothing in the game appeared to read. It reads it constantly: the dual is spanned by the bit-planes of the Grundy values — the parity checks are the value table stood on end — and on Mock Turtles over eight coins the losing rows are exactly the span of the table that decides them.
Impartial games
One proof, and one wrong lemma
Two measured identities were left for a proof: the move rule by induction, the gap condition from the reply bound. The induction is exact on 31,731 heaps at eight factors. The reply bound holds at c = 2 and on one index pair in twenty-seven at c = 3 — and the inequality that does the work is a third one nobody proposed.
Particular games
The obvious cut is the wrong one
Maundy Cake's rule was proved by restating its lemma so the short side vanished. Cutcake's collapses the same way — into a binary length instead of a multiset of primes — but the cut the argument needs is not the one the ladder predicted. Halving is wrong on a third of all cakes, and the smallest counterexample is six squares by two.
Particular games
The square that cannot be halved
Every number in hopless Toads and Frogs is a whole number, which the rung below measured on seven thousand strips and could not explain. The reason is that every empty square is either one player's alone or split evenly between them — except one, and that one is where the numbers stop.
Values
The bend above the top
The chain reading gets three values in 2,403 wrong because it counts bends that the diagram never reaches. Counting only the bends below the position's own temperature fixes all three and breaks none — the first exact reading on this ladder, and it needs one comparison rather than the envelope the rung below expected.
Values
A reduction that reads a graph
The two reductions are defined as deletions from an option list, and the shared form has no option lists — a node is reached from several parents at once. Both restate as rewritings at a node, the rewriting is confluent, and its fixed point is the canonical form. What does not carry over is the sharing: four fifths of the shared nodes need a different answer under different parents.