MathDB

Bowling Pins is a game played between two players in the following way: We start with

14

bowling pins in a line:
X X X X X X X X X X X X X X
Players alternate turns. On each turn, the player can knock down one, two or three consecutive pins at a time. For example:
Jing Jing bowls:
X X \:\:\:\: X X X X X X X X X X
Soumya bowls:
X X \:\:\:\: X X X X X X X X X
Jing Jing bowls again:
X X \:\:\:\: X X X X X \:\: X
The player who knocks down the last pin wins. In the above game, it is Soumya’s turn. If he plays perfectly from here, he has a winning strategy (In fact, he has four diﬀerent winning moves.) Imagine it’s Jing Jing’s turn to play and the game looks as follows
X \:\: X\dots
with 1 X on the left and a string of

k

consecutive X’s on the right. For what values of

k

from

1

10

does she have a winning strategy?

Problem Statement