MathDB

Anton and Britta play a game with the set

M=\left \{ 1,2,\dots,n-1 \right \}

where

n \geq 5

is an odd integer. In each step Anton removes a number from

M

and puts it in his set

A

, and Britta removes a number from

M

and puts it in her set

B

(both

A

and

B

are empty to begin with). When

M

is empty, Anton picks two distinct numbers

x_1, x_2

from

A

and shows them to Britta. Britta then picks two distinct numbers

y_1, y_2

from

B

. Britta wins if

(x_1x_2(x_1-y_1)(x_2-y_2))^{\frac{n-1}{2}}\equiv 1\mod n

otherwise Anton wins. Find all

n

for which Britta has a winning strategy.

Problem Statement