256 
[672 
672. 
ON THE GAME OF MOUSETRAP. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), 
pp. 8—10.] 
In the note “A Problem in Permutations,” Quarterly Mathematical Journal, t. I. 
(1857), p. 79, [161], I have spoken of the problem of permutations presented by this 
game. 
A set of cards—ace, two, three, &c., say up to thirteen—are arranged (in any order) 
in a circle with their faces upwards; you begin at any card, and count one, two, 
three, &c., and if upon counting, suppose the number five, you arrive at the card 
five, the card is thrown out; and beginning again with the next card, you count 
one, two, three, &c., throwing out (if the case happen) a new card as before, and so 
on until you have counted up to thirteen, without coming to a card which has to 
be thrown out. The original question proposed was: for any given number of cards 
to find the arrangement (if any) which would throw out all the cards in a given 
order; but (instead of this) we may consider all the different arrangements of the 
cards, and inquire how many of these there are in which all or any given smaller 
number of the cards will be thrown out; and (in the several cases) in what orders 
the cards are thrown out. Thus to take the simple case of four cards, the different 
arrangements, with the cards thrown out in each, are