Here is a 2-player game played on a region of the
Gaussian integers, $\mathbb{Z}[i]$.
Initially four points are colored, opposite
corners of an $X$ by $Y$ rectangle:
$0 + 0i$ and $X + Yi$ are colored red,
and $0 + Yi$ and $X + 0i$ are colored blue.
A move by a player of color $C$ consists of selecting a red point and a blue
point, and coloring the previously uncolored "midpoint" color $C$,
where the *midpoint* of $z+w$ is
$\lfloor{ (z+w)/2 } \rfloor$.
The game ends when a player loses by coloring a [Gaussian prime][1],
that is, a point $a + bi$, which, if either $a$ or $b$ is zero,
is a prime of the form $4n+3$, or otherwise if its norm $a^2+b^2$ is a prime.

<b>Example.</b> $X=51$, $Y=34$.
Red moves first and colors $25 + 0i$ red, the midpoint of
the points on the real axis.
Blue would not want to color the midpoint of $0 + 34i$ and $25 + 0i$,
because that is $12 + 17i$ which is prime (433).
So suppose Blue instead colors the midpoint of the points
on the imaginary axis, $0 + 0i$ and $0 + 34i$,
which is $0 +17i$.
Red could now select the midpoint of this blue point and $25 + 0i$,
which is $12 + 8i$.
And so on.

Let $X \ge Y$.
If $Y=0$, say that the game starts with just two points colored.
Then
the game could end
in a forced draw, e.g., with $X=8$: $(4, 6, 5)$.
Assume $Y > 0$.  Will the game always end with a winner?

If both $X$ and $Y$ are of the form $2p$ or $2p+1$ with $p$
a prime $4n+3$, then Red is forced to lose on the first move.
[Sorry if you read the first version of the preceding sentence, which was incorrectly stated.]
Are there other values of $X$ and $Y$ for which the game
can be fully analyzed?

Is there any hope
analyzing who wins this game (under best play) for arbitrary
$X$ and $Y$?

This is original and quite possibly worthless, so *caveat lector*!


  [1]: http://en.wikipedia.org/wiki/Gaussian_prime#As_a_unique_factorization_domain