A Problem in Permutations: the Game of 'Mousetrap'

Abstract

The game of 'Mousetrap, a problem in permutations, first introduced by Arthur Cayley in 1857 and independently addressed by Cayley and Adolph Steen in 1878, has been largely unexamined since. The game involves permutations of n cards numbered consecutively from 1 to n. The cards are laid out face up in some order and the game is played by counting on the cards, beginning the count with 1. If at any time the number of the count matches the number on the card, this is called a hit and the card is thrown out. The counting begins again with 1 on the next card and returns to the first card when the nth card is reached. Each time a card is hit, that card is removed and the counting starts over at 1. The game continues until all the cards have been hit and thrown out (the player wins) or until the counting reaches n with no cards having been hit (the cards win). The game is re-introduced here and a summary of both Cayley's and Steen's work is presented. Computer programs, written to generate the types of permutations dealt with by Steen, uncovered discrepancies in his work. Further examination of these discrepancies lead to the discovery of a combinatorial pattern of coefficients which Steen was unable to recognize because of his computational errors. Corrected versions of Steen's erroneous formulas are presented.