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 5 of 6: The position 6 reduces to 30

Previous step Next step

n_0=6:\quad n_1=30\text{ is A's only nonlosing move}

Detailed analysis

For n0=6, choices through 29 lose by the preceding bounds. If A chooses 31,32,33,34,35, or 36, B can choose respectively 1,1,3,2,5,4. Thus A's only nonlosing move is 30. From 30, B can choose 6,10, or 15; choosing 10 or 15 loses by the preceding analysis, so optimal B chooses 6.