Mathematical_Recreations-Kraitchik-2e

Games '295
end of the game. When all the pieces of one player have been
killed or captured, he retires from the game, his pieces are
removed from the board and the remaining two players continue
as in the game for two.
14. THE MOUSE TRAP. n numbered objects are placed
in a circle in any order. A player begins at anyone of them
and counts in order, "1, 2, 3, ...," until either the number
he speaks coi!lcides with the number of the object counted,
or until he has exhausted the n numbers. In the former case
(called a success), he withdraws from the circle the piece on
which the success occurred, and starts counting from 1 again,
beginning at the next object in the circle. He continues thus
until he has either failed to obtain a success or has removed
all the pieces from the circle. The player wins only if aLL the
objects are removed.
With four things in the order 3214 the first coincidence will
occur on 2, and the second on 1, but no further coincidences
can happen. With the order 1423 the player will win, taking
the pieces away in the order 1234. 8imilarly, 5 pieces arranged
in the order 13254 can be taken away in the order
12345.
Cayley has shown that there are 9 arrangements of 4 things
that give no coincidences, 7 that give only 1, 3 that give 2,
and 5 that give 4.
15. THE (FRENCH) POLYTECHNIC SCHOOL'S GAME. Starting
with a square array of (n + 1)2 dots the two players alternately
draw horizontal or vertical lines joining pairs of
adjacent dots. When a player draws the fourth side of one
of the n 2 small squares which may be formed in the figure,
he initials the square as credited to him. The player who
completes the greater number of such squares is the winner.
Usually the rule is made that a player who completes a
square may draw another line in the same turn, so that a