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  • Print publication year: 2009
  • Online publication date: February 2011

Monotonic sequence games


Abstract. In a monotonic sequence game, two players alternately choose elements of a sequence from some fixed ordered set. The game ends when the resulting sequence contains either an ascending subsequence of length a or a descending one of length d. We investigate the behaviour of this game when played on finite linear orders or ℚ and provide some general observations for play on arbitrary ordered sets.


Monotonic sequence games were introduced by Harary, Sagan and West in. We paraphrase the description of the rules as follows:

From a deck of cards labelled with the integers from 1 through n, two players take turns choosing a card and adding it to the right hand end of a row of cards. The game ends when there is a subsequence of a cards in the row whose values form an ascending sequence, or of d cards whose values form a descending sequence.

The parameters a, d, and n are set before the game begins. There are two possible methods for determining the winner of the game. In the normal form of the game, the winner is the player who places the last card (which forms an ascending or descending sequence of the required length). In the misère form of the game, that player is the loser. In these are called the achievement and avoidance forms of the game respectively.