Cayley's mousetrap

Mousetrap is the name of a game introduced by the English mathematician Arthur Cayley. In the game, cards numbered $1$ through $n$ ("say thirteen" in Cayley's original article) are shuffled to place them in some random permutation and are arranged in a circle with their faces up. Then, starting with the first card, the player begins counting $1,2,3,...$ and moving to the next card as the count is incremented. If at any point the player's current count matches the number on the card currently being pointed to, that card is removed from the circle and the player starts all over at $1$ on the next card. If the player ever removes all of the cards from the permutation in this manner, then the player wins. If the player reaches the count $n+1$ and cards still remain, then the game is lost.

The number of ways the cards can be arranged such that at least one card is in the proper place for n = 1, 2, ..., are 1, 1, 4, 15, 76, 455, ... This is sequence OEIS: A002467 in the Online Encyclopedia of Integer Sequences.

References

• Cayley, Arthur (1878), "On the game of Mousetrap", Quarterly Journal of Pure and Applied Mathematics, 15: 8–10. University of Göttingen Göttinger Digitalisierungszentrum (GDZ) scan
• Guy, Richard K.; Nowakowski, Richard J. (1993), "Mousetrap", in Miklos, D.; Sos, V. T.; Szonyi, T. (eds.), Combinatorics, Paul Erdős is Eighty, Bolyai Society Math. Studies, vol. 1, pp. 193–206, MR 1249712.
• Mundfrom, Daniel J. (1994), "A problem in permutations: the game of 'Mousetrap'", European Journal of Combinatorics, 15 (6): 555–560, doi:10.1006/eujc.1994.1057, MR 1302079.
• Spivey, Michael Z. (2009), "Staircase rook polynomials and Cayley's game of Mousetrap" (PDF), European Journal of Combinatorics, 30 (2): 532–539, doi:10.1016/j.ejc.2008.04.005, MR 2489284.

External links

• Weisstein, Eric W. "Mousetrap". MathWorld.