Minimal Blocks

Suppose we define an ordering on the points of the plane by saying that (x₁,y₁) ≤ (x₂,y₂) if and only if x₁ ≤ x₂ and y₁ ≤ y₂. The ordering is only partial, i.e., doesn't apply to all pairs of points, but then when it does apply we can probably all agree with the result.

Under that ordering, a set of points won't necessarily have a minimum, but it will have a minimal set, a set of points such that every other point is larger than at least one of them. Here's the minimal set for the points that are centers of 3×3 blocks.

Actually, that's only half the minimal set … you can get the other half by reflecting about the line y = x. Next, here are the coordinates, reflected to fit the convention that x < y.

(	105	,	6201	)
	231		5655
	495		5301
	595		3129
	1275		1309

Which of these is the “real” first block is a matter of taste. However … if we measure using the function min(x,y), the first block is the first one I found, but under any other reasonable measure—max(x,y), x+y, sqrt(x²+y²)—the first block is (1275,1309).

Finally, here are three nice non-minimal blocks that I happened to notice.

(	1001	,	3795	)
	2001		1885
	1729		7125