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 1 of 6: A forces 1990 from all n at least 8

Previous step Next step

8\le n\le11\mapsto60,\quad12\le n\le16\mapsto140,\quad17\le n\le22\mapsto280,\quad23\le n\le44\mapsto504,\quad45\le n\le1990\mapsto1990

Detailed analysis

When A receives n, use $8\le n\le11\mapsto60$ , $12\le n\le16\mapsto140$ , $17\le n\le22\mapsto280$ , $23\le n\le44\mapsto504$ , and $45\le n\le1990\mapsto1990$ . Each target lies between n and n^2, so it is legal.