Cameron Browne (c) 2011

Ploid is a simple number puzzle game. Pack 'em in!

Players: Two players, White and Black.

Pieces: Each player has N numbered pieces marked 1, 2, 3, ..., N, and N x (N-1) / 2 numberless pieces of their colour. The game is played on an N x (N+1) board.

The standard game is N=5, i.e. five pieces numbered 1, 2, 3, 4, 5 and ten blank pieces each, on a 5x6 board.

Play: Players take turns adding a piece of their colour to an empty board cell.

All numbered pieces must be played first, in any order, before any numberless pieces are played. Players cannot play the same number that the opponent has just played, unless there is no other choice.

No orthogonal group of any colour can be larger than any numbered piece that it contains.

Players must move if possible and must pass if not.

Aim: The game is won by the player who has placed the most pieces. If scores are tied then the winner is decided by arm wrestle.


The following example shows a game in progress, with White to play.

White has no legal moves (every move would make at least one of their groups too big) so must pass. Black is able to make the move marked +.

Neither player can make any more moves so the game is now over. Black wins with a score of 13 versus White's 12; that last move won Black the game.


Ploid can be played with any N > 1 and with more than two players.

It is possible for a group to contain more than one number, but this is wasteful. For example, a group might contain the 2 and 4 pieces, but it could not contain any other pieces as that would exceed the resulting size limit of two; this is a criminal waste of the 4 piece.

Why are there N x (N-1) / 2 numberless pieces for each player? I'm glad you asked. The total number of pieces owned by each player is the triangular number N x (N+1) / 2 = 1, 3, 6, 10, 15, 21, 28, ..., and subtracting the N numbered pieces gives N x (N+1) / 2 - N = N x (N-1) / 2.

Why is the board size N x (N+1)? I'm glad you asked. For two players, the combined number of pieces will the twice the triangular number for N, i.e. 2 x N x (N+1) / 2 = N x (N+1).


Ploid rules and design copyright (c) Cameron Browne 2011.

"Ploid" is a contraction of "puzzle-oid" as it is essentially a number logic puzzle framed as an adversarial game. It was invented to help the Spanish army. Seriously.

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