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 2 of 6: B's replies to the forcing targets

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504\to\{56,63,72,168\},\quad280\to\{35,40,56,70,140\},\quad140\to\{20,28,35,70\},\quad60\to\{12,15,20,30\}

Detailed analysis

The B replies that avoid an immediate A win from these targets are exactly $504\to\{56,63,72,168\}$ , $280\to\{35,40,56,70,140\}$ , $140\to\{20,28,35,70\}$ , and $60\to\{12,15,20,30\}$ . Any omitted reply already lies in the range $45\le n\le1990$ and lets A choose 1990. Every listed result lies in the next interval in A's strategy, so A reaches 1990 after finitely many rounds.