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,
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.

**Example.** $X=51$, $Y=34$.
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.

Will the game always end with a loser?

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. 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*!

**Edit1.** See the suggested simplification by Michael Albert in the Comments: dispense with
colors, letting each move select any two points.

**Edit2.** Thanks for all the interesting comments.
It now seems to me this game is hopelessly complicated to analyze, perhaps PSPACE-complete in terms of complexity. The monochromatic version is much simpler but removes the
adversarial aspect that is the essence of a game. I don't think Milton-Bradley will be knocking on my door!

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