Question 45
Let $n \ge 2$ be a positive integer. A square with the side length $2n - 1$ is divided into $(2n - 1)^2$ unit squares using parallels to the sides. Ana and Bogdan play the following game: starting with Ana, the two take turns to colour, Ana with red, Bogdan with blue, in $2n^2$ turns, the $4n^2$ vertices of the unit squares. Then, starting with Ana, each of them connects with a vector a red dot (which is the tail of the vector) with a blue dot (which is the head of the vector), obtaining $2n^2$ vectors with distinct tails and heads. If the sum of these vectors is zero, Ana wins. Otherwise, Bogdan wins. Show that Bogdan has a winning strategy.
Cristian Șăvescu
Cristian Șăvescu