Problem 5

Given

n_0>1

, players A and B choose integers alternately. Knowing

n_{2k}

, A chooses

n_{2k+1}

with

n_{2k}\le n_{2k+1}\le n_{2k}^2

. Knowing

n_{2k+1}

, B chooses

n_{2k+2}

such that

n_{2k+1}/n_{2k+2}=p^r

for a prime

p

and integer

r\ge1

. A wins by choosing

1990

, and B wins by choosing

1

. Classify the initial values according to which player has a winning strategy or neither does.

Step 4 of 6: B wins from 2 through 5

Previous step Next step

n_0=2,3,4,5\Longrightarrow\text{B wins}

Detailed analysis

If B receives a number below 6, B chooses 1. If B receives a number at most 11, B chooses 1 or 2; at most 19, chooses 1,2, or 3; at most 29, chooses 1,2,3, or 4. These bounds show that after A starts with n0=2,3,4,5, every legal A move gives B an immediate winning reply.