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 6 of 6: The position 7 also reduces to 6

Previous step Next step

n_0=7:\quad n_1\in\{30,42\},\qquad30\to6,\quad42\to6

Detailed analysis

For n0=7, all choices except 30 and 42 have explicit B replies: choices 7 through 29 lose by the bounds, and choices 31 through 49 (apart from 42) are defeated by the corresponding small divisors listed in the source strategy. From 42, B can choose 6,14, or 21; 14 and 21 are losing for B, so B chooses 6. Thus both 6 and 7 are draw positions, while the classifications are: A wins iff n0≥8, B wins for n0=2,3,4,5, and neither wins for n0=6,7.