Combinatorial Game Theory
who wins, and by how much
Generator

Shove strips, and what each is worth

Shove strips, and what each is worth
Shove strips, and what each is worth. A shelf of positions with the value the recursion returns beside each. Every one is a number: Shove has no hot positions at all, which is unusual for a partizan game and is the first of the essay's three claims.

A shelf of positions with the value the recursion returns beside each. Every one is a number: Shove has no hot positions at all, which is unusual for a partizan game and is the first of the essay's three claims.

12 essays call shove-row. 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

44 distinct positions, harvested by running this generator again at the options each essay passed it.

PositionWorth OutcomeDrawn in
...LLLR −73/512 R The criterion that cannot exist
...LLR −21/64 R The other way to move a row
...RRL 21/64 L The other way to move a row
..LL.R −3/8 R The reading that survives too much
..LLLR −9/64 R The criterion that cannot exist · The other way to move a row
..RR.L 3/8 L The reading that survives too much
..RRRL 9/64 L The other way to move a row
.L 1 L Nothing worth fighting over · The other way to move a row · The reading that survives too much
.L.L.R −1/2 R The reading that survives too much
.LL.R −1/2 R The reading that survives too much
.R −1 R The other way to move a row
.RR.L 1/2 L The reading that survives too much
L 0 P Nothing worth fighting over · The other way to move a row · The reading that survives too much
L.R −1 R Nothing worth fighting over · The other way to move a row · The reading that survives too much
LL 0 P The other way to move a row
LR 0 P Nothing worth fighting over · The other way to move a row · The reading that survives too much
R 0 P The other way to move a row
R.LL 2 L The other way to move a row · The reading that survives too much
RL.L 1 L The other way to move a row · The reading that survives too much
RLL 0 P The other way to move a row · The reading that survives too much
..L 3 L Below zero · Nothing worth fighting over · The other way to move a row
..LRL 15/4 L Nobody comes back · What the class does not buy
..R −3 R Below zero · Nothing worth fighting over · The other way to move a row
.LL 5 L Below zero · Nothing worth fighting over · The other way to move a row
.LR −2 R Below zero · Nothing worth fighting over · The other way to move a row
.RL 2 L Nothing worth fighting over
L 1 L Nothing worth fighting over
L.L 4 L Nothing worth fighting over · The reading that survives too much
L.R −5/2 R Nothing worth fighting over · The reading that survives too much
LL 3 L Nothing worth fighting over · The reading that survives too much
LLL 6 L Nothing worth fighting over · The reading that survives too much
LLLL 10 L Nothing worth fighting over
LLLR −15/8 R Nothing worth fighting over
LLR −7/4 R Nothing worth fighting over · The reading that survives too much
LLRR −23/4 R Nobody comes back · Nothing worth fighting over · What the class does not buy
LR −3/2 R Nothing worth fighting over · The reading that survives too much
LRL 9/4 L Nothing worth fighting over
LRLR −23/8 R Nobody comes back · Nothing worth fighting over · What the class does not buy
LRR −9/2 R Nothing worth fighting over · The reading that survives too much
LRRR −17/2 R Nothing worth fighting over
RL 3/2 L Nothing worth fighting over
RRL 7/4 L Nothing worth fighting over
RRRL 15/8 L Nothing worth fighting over
RRRRL 31/16 L Nothing worth fighting over

Where it is called

Changing this generator changes every one of these figures.

Shove strips, and what each is worth. A shelf of positions with the value the recursion returns beside each. Every one is a number: Shove has no hot positions at all, which is unusual for a partizan game and is the first of the essay's three claims. Particular games

Nothing worth fighting over

Shove is a strip of coins beside a cliff, and both players have completely different moves. Every one of its 728 positions is worth a number, so nobody ever wants to move; the winner is the owner of the coin furthest from the cliff, in all 728; and the number the board is worth is not the sum of its coins — that reading is exact on 126 strips and wrong on 588 of the other 602.

 Where 1,474 values sit on the scale. The temperature of every value in the pool, counted. The floor is −1 and only numbers are on it; the next rung up is 0, and everything there is a number with an infinitesimal added. Above that the scale is continuous and the counts thin out. Temperature

Below zero

Temperature is described as urgency and urgency has no obvious bottom, but the scale stops at −1 and only the numbers are on it. The rung above is exactly zero, and the 337 values born by day three that sit there are the ones a number cannot be told from — flat thermograph, nothing at stake, and an infinitesimal that no number can see. Two independent computations agree on the classification for all 1,474.

 Push and Shove over every strip up to 6 squares. The same strips under both rules. A cliff lets coins fall off and a wall does not, and the census says what that one clause is worth: both games are entirely made of numbers, they never agree on a value, and the obvious board-reading is right far more often under the wall than under the cliff. Particular games

The other way to move a row

Shove has a cliff and Push has a wall, and that is the whole of the difference. Both games make every one of the 728 strips up to six squares a number, so neither ever has anything worth fighting over — and the two rules do not agree on the value of a single position. The obvious board-reading is exact on 446 strips under the wall and on 140 under the cliff, and 486 strips contain a coin its owner cannot move at all.

 Where running out of moves is permanent. Eleven rulesets, each walked position by position from three small boards, with every position at which a player has no move examined for whether any continuation gives them one back. Nothing here is evaluated: dead-ending is a property of the rules, and two boards worth the same value can differ on it. 9 of the 11 are dead-ending and 2 are not. Where it stops

Nobody comes back

There is a class of games in which running out of moves is permanent, and it is the setting almost every modern misère result is stated in. Nine of this site's eleven rulesets belong to it across 5,334 positions; the two that do not are Toads and Frogs and Amazons, and Toads and Frogs loses the property to a single clause — delete the hop and it joins the list.

 When counting the free squares gets Push right. Every Push strip of at most seven squares, split by whether any line of play can bring two coins of opposite colour together. Where none can, the count of free squares in front of each coin is the value, without exception; where one can, the count is right more often than not. Particular games

The reading that survives too much

Counting the empty squares in front of each coin gets a Push position right half the time, and the rung below said the failures were exactly the positions with two coins of opposite colour side by side. Sixty-six of the 1,072 failures have no such pair, the smallest is five squares long, and the condition that does decide it is not about the board at all — it is about every position the board can reach.

 Three strips a criterion cannot tell apart. Three Push strips identical in length, reading, coin counts and run structure, whose readings are wrong by a quarter, a half and a quarter more than one. The order of the colours inside the run is the only thing separating them. Particular games

The criterion that cannot exist

The rung below asked for a quantitative version of its condition — turn 'the reading survives mixing three quarters of the time' into a statement about the strip. Three strips of four squares settle it. `.LLR`, `.LRL` and `.RLL` have the same length, the same reading, the same coins and the same single run, and their readings are wrong by 1¼, ¼ and ½. The error is a fact about the order of the colours, and 207 of 805 statistical classes carry more than one of them.

 How often one position beats another. Misère comparison inside each ruleset's own universe. A quarter to a half of ordered pairs compare, and the ruleset that is not dead-ending is in the middle of the range. Where it stops

What the class does not buy

Dead-ending is the hypothesis several modern misère results are stated under, and the rung below sorted this site's games into it without running the comparison those results are about. Running it: a quarter to a half of ordered pairs compare inside a ruleset's own universe, which is a great deal — and the ruleset that is not dead-ending sits in the middle of that range. Ten comparisons are lost when a universe is enlarged, and every one is lost to a dead-ending company.

 A numeral in the empty squares. Runs of coins with one to five empty squares in front, and the value of each. Every row is a binary expansion converging on a fraction the colours determine. 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.

 Moving them apart does not make them independent. The value of a two-run Push strip as the gap between the runs widens. Each row converges, and none of them converges to the sum of its two runs. 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.

 Every gap dies at the last run's rate. Three-run Push strips with each gap widened in turn, and which of four candidate rates the convergence matches. The rearmost run's rate wins every family and the compound rate wins none. Particular games

Read from the back forwards

The rung below found a two-run Push strip converging at a rate set by the back run and asked what a third run does — whether the rate is still the rearmost run's, or whether the rates compound. It is the rearmost run's, and for every gap: widen the front gap of a three-run strip, two whole runs away, and the value still dies at the last run's rate. Shove, the game one clause away, compounds.

 Hold the tail, vary everything in front. The grouping test: every tail against every prefix, asking whether the rate at which a widening gap stops mattering is a function of the tail alone. 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.

 The threshold is one whole move. The prefixes grouped by their own value, against the rate they produce behind LRRL. No value class splits, and the boundary falls exactly at one move: three quarters of a move is not enough and one move is, with nothing between them. A fractional advantage does not reach across a gap that grows without bound and a whole move does. Particular games

A fraction does not reach

Two Push tails read their prefix when every other tail ignores it, and the previous rung guessed the deciding bit was a shape — whether the prefix's last coin stands alone. The full census says it is a number. The rate changes exactly when the prefix is worth a whole move, and three quarters of a move is not enough.

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