Transposition

 Overview Transposition (also known as Odds vs. Evens) is a two player number game invented in 2000 by Gregory Keith Van Patten ().

 Cards Transposition game uses an even number of cards numbered 1 through N, where N is at least 8. The game begins with the cards arranged in increasing order, from left to right:

 Objective The first player wins if the odd numbered cards are arranged in decreasing order from left to right, but not necessarily consecutively. The second player wins if the even numbered cards are arranged in the same manner. It doesn't matter which player actually arranges the cards in the required order; if, after some move, the odd numbered cards are arranged in the required order, then the first player wins, even if the cards were arranged by the second player. If one move leads to a situation where both players' victory conditions are simultaneously satisfied, then the game ends in a tie.

 Play Players alternate moves. Passing is not allowed. On his turn, a player must switch any pair of cards, regardless of whether they are adjacent. The only constraint is that after the switch is made, the smaller card must be to the right of the larger one. Seen from another way, a pair may be switched if, before the switch is made, the smaller card is to the left. The pie rule is used for making the game fair. This rule allows the second player to switch places with the first player after the first player makes his first move, if he thinks that the first player has made an especially good move.

 Variants A pair may be swapped only if the cards are adjacent. Instead of odds and evens, the cards are divided into two equal groups based on their value, with the cards above N / 2 in one group, where N is the total number of cards. In this variant, the first player is trying to sort the half with lower values.