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 no other game on this site has.
5 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 running this generator again at the options each essay passed it.
| 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
Changing this generator changes every one of these figures.
Where it stops
A move that must be answered
Every argument on this site 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 on this site 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.
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