Chapter 1.1, Problem 4MC

Consider a game where you have two distict piles of objects.

Two players alternates moves, each player taking any number $(\text{not}0)$ of objects desired from a single pile.

The player who takes the last objects (so nothing is left in either pile) is the winner.

If there are $a$ objects in one pile and $b$ objects in the second, we write $(a,b)$.

Given the game $(1,1)$, the first player loses because if she takes $1$ from a pile, the second player takes $1$ from the other pile. Answer the following:

a. Show that the first player can win the game $(1,2)$ as well as $(1,100).$

b. Who will win the game $(1,a)$ if $a>1$? Why?

c. Which player will win the games $(2,2)$, $(3,3)$, $(4,4)$?

Why?

d. Which games can the first player always win? Why?