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The Next Block
 > Minimal Blocks
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## Minimal Blocks

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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

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