Combinatorial Game Theory
who wins, and by how much
Concept

Amazons — where it appears

A game whose queens fire arrows that burn squares, so that the board eventually cuts itself into regions no piece can cross. The split arrives during play rather than being assumed, so the sum is produced by the players rather than by the rules.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

A board in pieces costs the sum, not the product. A Domineering board with squares blocked out, so that it falls into regions no domino can span. The number of positions in the whole board is exactly the product of the numbers in its regions — which is why evaluating the regions separately, and adding the values, is an exponential saving rather than a tidier way of writing the same search.

The board falls apart, and the arithmetic changes

A 4×5 Domineering board with a wall down the middle has 2,916 positions in it, and that number is exactly 54 × 54 — the product of its two halves. Solving the halves separately costs 108. Decomposition is the one saving in this subject that turns a product into a sum.

complexity · Decomposition Amazons, after the arrows have cut the board in 2. An amazon moves like a queen and then shoots an arrow, also like a queen, which burns the square it lands on. Late in a game the burnt squares cut the board into regions no amazon can cross — and from that moment the position is a sum of independent games, which is the shape the whole theory was built for, arrived at by the play rather than assumed.

Amazons, and when a position becomes a sum

Every technique on this site starts from a position already broken into independent parts. Amazons does not begin that way — the board is one fight until the arrows cut it, and the moment of cutting is something the play produces rather than the analyst assumes.

positions · Amazons Where the count and the value part company. Amazons endgames whose arrows have already cut the board into regions, with the territory count beside the computed value. Territory gives every empty square to whichever amazon can reach it in fewer moves, which is what Amazons programs compute. The positions drawn are the ones where that number gets the outcome wrong, and they have something in common: each is worth a switch, so there is no number for the count to have been right about.

When a real board falls apart

Amazons is played competitively, and late in a game the arrows have cut the board into regions no piece can cross. From that moment the position is a disjunctive sum — arrived at by the play rather than assumed — and the territory count every program uses can be measured against what the sum is actually worth.

applied · Amazons Amazons on one line. A one-dimensional Amazons board: an amazon slides along the row and shoots along the row, and the square the arrow lands on is burnt for the rest of the game. The whole board fits in a sentence, and the values it produces are already of several different kinds.

Amazons on one line

A board one square high is small enough to evaluate completely: every strip from two to ten squares with one amazon a side is 37,886 positions taking 81 distinct values, and every one of them is an integer, a switch, a number plus a star, or a bare star. Not one is a fraction — and forcing the arrow onto the square just vacated, which takes a freedom away rather than adding one, produces 1,196 that are.

positions · Amazons 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.

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.

limits · Dead-ending Still a count of squares. One-sided Amazons regions on two-dimensional boards, swept exhaustively. Every one is worth exactly the number of free squares in it.

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.

positions · Amazons Almost none of them is a number. Every three by three Amazons region holding one amazon of each colour, sorted by what kind of value it carries. Fifty-six of the two thousand are fractions and the great majority are hot positions.

The fractions that were not there

The rung below counted 1,452 fractions among the shared Amazons regions and asked which fractions they are. Fifty-six of them are fractions. The other 1,396 are hot positions with a fraction somewhere inside their options, counted by a regular expression looking for a slash — and the quantity the separation of the two amazons actually sets is not a denominator but a temperature.

positions · Amazons The same distance, different room. Two shared Amazons regions with the amazons the same distance apart, differing in how many squares both of them can still reach. The one sharing more is the colder.

Room pulls two ways

The rung below found the distance between two amazons setting a shared region's temperature and asked for something finer — the squares each can reach, or the squares both can. Neither beats the distance on its own. Together they beat it by half as much again, and the reason is that they pull opposite ways: further apart is hotter, and sharing more reachable squares is colder.

positions · Amazons What each stage buys. The account built up one quantity at a time, with the random control priced beside the last row.

An effect that changes sign

Which squares two Amazons share turns out to matter about as much as how many — three shared squares in a line run at 0.63 where three scattered run at 2.51. But the effect of clumping is hotter at one distance and colder at the next, so the arrangement predicts well and describes nothing, which is not what the four rungs below it produced.

positions · Amazons One board, two answers to how many pieces it is in. Every position reachable from a small Amazons opening, counted by depth, under two ways of deciding whether two squares are in the same region. Counting only edge neighbours, a third of all positions are in pieces; counting corners too, an eighth are.

A wall an amazon can walk through

An arrow burns a square for good, so an Amazons board that has fallen into pieces should stay in pieces. Over 127,583 positions it does not: fifty-one thousand moves put two regions back together. Every one of them is a single diagonal step, and what is wrong is not the game but the rule used to find the regions — which was borrowed from a game whose pieces lie along the board's own lines.

complexity · Decomposition The third point. Amazons regions on a 4 × 4 board holding one amazon of each colour, by the Chebyshev distance between them. A 3 × 3 board reaches distance two and this one reaches three, and the mean temperature rises across all three.

The third point on the curve

Four rungs below this one measure a shared Amazons region's temperature against how far apart its two amazons are, and every one does it on a 3 × 3 board, where the distance can only be 1 or 2. Two points make a direction, not a curve. A 4 × 4 board reaches distance 3 — and cannot be evaluated at all until the regions are cut to five free squares. Restricted that far, the answer is neither a sign that flips nor an oscillation: the rise continues and it is running out, the second step being 36 per cent of the first.

positions · Amazons

Named alongside it

The objects these essays reach for when they reach for this one.

DecompositionRegionEnumerationExhaustive searchHeuristicTemperatureValueDisjunctive sumInvariantPartizanApproximationComponent

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