Top Entails, one heap at a time
Each heap with the outcome of playing it alone, the Grundy value an ordinary solver would give it, and the moves that win from it. Taking the top coin of a heap forces the opponent to answer in that heap, which is a kind of move an ordinary sum of games has no room for.
8 essays call
entail-heaps. 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
13 distinct positions, harvested by drawing the figure again at the options each essay asked for.
| Position | Worth | Outcome | Drawn in |
|---|---|---|---|
a Top Entails heap of 1 |
no nimber |
N | A move that must be answered · A pass is not a move · What a component has to carry |
a Top Entails heap of 2 |
no nimber |
N | A move that must be answered · A pass is not a move · What a component has to carry |
a Top Entails heap of 3 |
no nimber |
P | A move that must be answered · A pass is not a move · What a component has to carry |
a Top Entails heap of 4 |
no nimber |
N | A move that must be answered · A pass is not a move · What a component has to carry |
a Top Entails heap of 6 |
no nimber |
N | A move that must be answered |
a Top Entails heap of 8 |
no nimber |
N | A move that must be answered |
Top Entails heaps of 2 and 2 |
no nimber |
N | A move that must be answered |
Top Entails heaps of 2 and 4 |
no nimber |
N | A move that must be answered |
Top Entails heaps of 2 and 6 |
no nimber |
N | A move that must be answered |
Top Entails heaps of 4 and 2 |
no nimber |
N | A move that must be answered |
two Top Entails heaps of 2 |
no nimber |
N | A move that must be answered |
two Top Entails heaps of 4 |
no nimber |
N | A move that must be answered |
two Top Entails heaps of 6 |
no nimber |
N | A move that must be answered |
Where it is called
Every one of these figures is a drawing of the same construction, so a correction to it corrects them all.
Where it stops
A move that must be answered
Every argument about sums assumes the parts are independent: a move in one leaves the others alone, and the reply may go anywhere. Top Entails denies it — take the top coin of a heap and the opponent must answer in that heap. The nim-sum then misreads 9 of 36 two-heap positions, and two heaps of two coins are a first-player win, which no impartial game the theory covers can be.
Where it stops
A pass is not a move
Put a single pass token on a Nim board and one clause decides everything. If it may be taken at any time — including as the move that ends the game — the value of the whole is the nim-sum with a one added, in all 120 positions swept: the pass is a heap of one. Forbid it as the final move and the value stops being a function of the nim-sum at all, and 3 and 1 + 2 come apart.
Where it stops
What a component has to carry
Three impartial games — Fibonacci Nim, Top Entails and Nim with a held pass — break the sum, and they break it for the same reason: a component cannot say what its own legal moves are. Measured with one instrument — one number per part, exclusive-ored — the failure rate runs from a quarter to nearly half, against a control where the same recipe is a theorem and is never wrong.
Where it stops
Three heaps and a pass
Nim with a single pass that may not end the game is easy on one heap and on two: a heap swaps each odd size with the even one above it, and two heaps lose exactly at (2k − 1, 2k). On three heaps the losses are known only as a list. Fix the smallest heap and each slice of the list settles into a pattern after an irregular start — period 4, 8, 10, then 160 at a smallest heap of ten, and nothing visible from eleven.
Where it stops
What a component would have to carry
For a held pass to be decided by a summary of each component, the summary must separate every pair of components some company tells apart. The Grundy value does not — Nim 1 and Kayles 8 are equal games that a held pass separates beside a single Nim heap of two. Nor does the Grundy value with the component's own held-pass value: Kayles 3 and Kayles 6 agree on both and are split by a company of two Nim heaps. Over twenty-four components, fifteen classes against fourteen pairs, and the gap widens as the pool grows.
Where it stops
One value more than Nim
Top Entails broke the nim-sum: nine of thirty-six two-heap positions came out wrong when a move could compel the reply. The repair is one extra value. Read each heap off the heap itself, and every heap of Top Entails and of a second entailing game is either a nimber or loony — a win for the mover in any company. One rule then decides every sum of those games and Nim together: 4,016 positions, and it is never wrong. A loony heap is a pass, and the compulsion is what buys it.
Where it stops
The table closes, until heaps of four
Classes that no company can tell apart always combine, so the question left by the held pass is not whether it has an algebra but whether the algebra is finite. Computed the way a misère quotient is, it is: four classes for heaps of one where misère play needs two, eleven for heaps up to two, fourteen up to three, nineteen for Kayles rows up to five. Admit Nim heaps of four and the table never closes — a single line of losses, 2k + 3 heaps of two beside 2k heaps of four, gives every count of twos a class of its own. And at the bottom of every closed table sits the normal-play group, with the loss moved to nim-sum one: on a crowded board the held pass is a heap of one.
Where it stops
A compulsion that points elsewhere
Top Entails and entailing Nim are decided by one value more than Nim because the compelled reply stays in the heap that compelled it. Point it at a different heap instead and the two games part company. Entailing Nim does not notice — its heaps are worth exactly what Nim's are, because Nim's winning reply was never in the heap just moved. Top Entails loses its loony heaps and then every value of the Nim kind: no assignment of nimbers explains its sums, because a heap of three beside two heaps of one is a win and beside two heaps of two is a loss.
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