Cayley's mousetrap
Mousetrap is the name of a game introduced by the English mathematician Arthur Cayley. In the game, cards numbered 
through 
("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 
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 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 
and cards still remain, then the game is lost.
In order for at least one card to be removed, the initial permutation of the cards must not be a derangement. However, this is not a sufficient condition for winning, because it does not take into account subsequent removals. The number of ways the cards can be arranged such that the entire game is won, for n = 1, 2, ..., are
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., Combinatorics, Paul Erdős is Eighty, Bolyai Society Math. Studies 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", European Journal of Combinatorics 30 (2): 532–539, doi:10.1016/j.ejc.2008.04.005, MR 2489284.