Discrete Mathematics Level 5

Two players play a game on the cartesian plane which occurs in 2 phases. In the first phase, a number \(n\) is selected from \(\{2,3,4,\ldots, 999\}.\) The first player then places a point at a location \((x,y)\) in the plane satisfying \(-n \leq x \leq n, -n \leq y \leq n\) and \(x,y\) integers. The players alternate placing points until a total of \(n\) points have been placed.

In the second phase, the first player picks two distinct points in the plane that are not joined and joins them by a curve that does not intersect itself, any of the other \(n-2\) points, or any of the already drawn curves. The players alternate turns, drawing curves in this manner. The first player who is unable to draw a curve loses. For the 998 starting values of \(n,\) determine how many of these the first player has a winning strategy for.

Details and assumptions

The players get to choose where they want to place their points.

The curve is a continuous path which does not include the endpoints. Hence, 2 curves may seem to intersect at one of the \(n\) points.


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