Game of the Amazons

How to play Amazons

The board is 10 by 10. Eight amazons stand on it at the start, four to a side, set in a ring around an empty middle: two on your back row, three squares in from each corner, and one against each side wall. Bronze opens from the near side and verdigris answers from the far side. The literature calls the two sides white and black.

A turn is two things and you must do both. Move one of your amazons any distance in a straight line, along a rank, a file or a diagonal, over empty squares only, the way a queen moves in chess. Then, from the square she has landed on, shoot an arrow down another line of the same kind. The arrow stops where you send it and burns that square for the rest of the game. Nothing moves onto a burned square, nothing can cross one, and neither player gets it back. Moving without shooting is not a turn.

The square your amazon left is empty by the time she shoots, so she is allowed to put the arrow straight back into it. That catches people the first time, and it has a consequence worth knowing: a move that leaves an amazon with nowhere to shoot cannot happen, because the line she arrived on is always open behind her. Having a legal move and having a legal turn are the same thing here.

The last player able to move wins, so the first player with no legal amazon move has lost. There is no pass. Draws are impossible, because every turn burns one more square, none ever come back, and a board that only shrinks cannot go on forever.

Invented in 1988, printed in 1992

Somebody designed that turn. Most of the boards here were dug up, or reconstructed from a text somebody wrote down long after the last person who knew the rules had died. This one has an inventor, a year and a first printing: a game invented in 1988 by Walter Zamkauskas of Argentina. The rules were printed in Spanish in the puzzle magazine El Acertijo in December 1992. Michael Keller’s approved English translation followed in World Game Review in January 1994, and the Spanish name, El Juego de las Amazonas, is a trademark of Ediciones de Mente. None of that is a sentence you can write about a board that came out of a tomb.

It looks like chess and it counts like Go

Nothing is captured, ever. The eight amazons that start the game are the eight that finish it, so the habit of counting material has nothing here to count. What changes hands is room. Each arrow takes one square away from both players at once. By the end the position is a comparison: how many moves you can still make inside your walls, against how many they can make inside theirs.

That puts the Game of the Amazons in the territory family rather than the capture family. The pieces move like chess. The counting, when you finally do it, is Go’s: whose space is bigger, measured in turns. It is not a connection game either, which is the other thing the board suggests. Joining two edges wins nothing here. The win comes from being the last side with a move left.

Why the Game of the Amazons is hard for computers

We counted the legal opening turns here, against the rules this table plays by. From the starting position the first player has 80 queen moves, each one opening its own fan of arrow shots, and the two multiply out to 2,176 distinct ways to take a first turn. Chess opens with 20. The Chess Programming Wiki puts the Amazons average across a whole game at 374 for the first player and 299 for the second.

A tree that wide brought the programmers in early. Amazons had its own event at the ICGA Computer Olympiad from London in 2000, eight years after the rules were printed, and the same handful of programs keep taking it. By 2017 Richard Lorentz’s Invader had eight Amazons gold medals to its name, by the Chess Programming Wiki’s count.

Generalized Amazons, the version played on an n by n board rather than a 10 by 10 one, is PSPACE-complete. Timothy Furtak, Masashi Kiyomi, Takeaki Uno and Michael Buro proved it at IJCAI in 2005, and Robert Hearn proved it separately the same year. Hard is not the same as solved. Nobody has solved the 10 by 10 game and we have found nothing claiming to, and we cannot tell you whether the best programs now beat the best people, because we have not found a match that settles it.

That width is the branching factor, and it does not last. We played a hundred games out against these same rules, picking at random from every complete legal turn: the count of legal turns dropped below a thousand at the twelfth turn on median, below a hundred by the forty second, and the games themselves ran between 49 and 83 turns, 72 on median. Random play is not skilled play, but the shape of the decay is the board’s doing rather than the player’s. A game that opens a hundred times wider than chess finishes with two or three things you are allowed to do.

The endgame is several small games at once

Arrows build walls and walls cut the board up, blocking whole regions off from each other. Some way into the middle game the position stops being one game and becomes a set of sealed rooms, each holding an amazon or two with no way in and no way out. A room is worth a fixed number of moves to whoever is standing in it, and the whole position is the sum of the rooms.

That property is what brought the mathematicians. Elwyn Berlekamp worked out combinatorial game values for sums of n by 2 Amazons, where each player keeps one amazon in each narrow strip, as a demonstration of what the theory can do with a real game. It is also how the strong programs win: solve the rooms one at a time instead of searching the whole board, because once the walls are solid the rooms cannot talk to each other.

Amazons strategy: territory is counted in turns

Since the game ends when somebody runs out of moves, the thing to count is not how much of the board looks like yours but how many turns you can still take inside it. A wide open region with one amazon in it can be worth less than a cramped one holding two.

Two mistakes cost new players most of their first games. An amazon walks deep into open space, an arrow closes the door behind her, and she spends the rest of the evening alone in a room she is emptying one square at a time. And the arrow gets treated as an afterthought when it is half of every turn and the only move that changes the board permanently.

The shot that usually pays is the one that splits. Put an arrow where it cuts a large shared space into two unequal halves, then make sure you have more amazons on the bigger side. Walling your opponent out is worth doing only when you have checked that it does not wall you in as well.

Playing the Game of the Amazons online, free

One click sets a table out and gives back a link. Whoever you send it to takes the far four amazons the moment they open it, from whatever room or country they happen to be sitting in, with no account asked for at either end and nothing to download. Two of you in the same place can work from a single screen instead, handing the device over after each arrow lands. With nobody else about, the computer will sit down on the far side.

The board shows you the squares you are allowed to use, first for the slide and then for the shot. That helps more here than in most games. An amazon has up to eight lines to travel along, and up to eight more to shoot down from wherever she stops, so the arithmetic is done for you while you think about the shape.

The Amazons board game, and where the name gets in the way

There is no single official box. The game was published as rules rather than as a product, and people have built their own ever since. A few makers sell wooden sets, none of them an official edition, and we have found none we could point you at as the standard one. The usual homemade version is a 10 by 10 draughts board, four queens a side borrowed from two chess sets, and a bowl of counters for the arrows. A standard chessboard is 8 by 8 and will not do. An international draughts board is 10 by 10 and will.

Going looking for that set is where the name starts working against the game. Type amazons into a search box and you meet the shop, the river and the women warriors of Greek myth well before you meet the board. A game from 1988 ends up exactly where the older boards here already are. You have to know what to ask for.