consider packing circles of diameter 1 into a square. the square lattice is less efficient than hexagonal close packing, but when does the difference begin to matter? we consider packings of n^2 circles to give the square lattice its best chance.

we rescale the coordinates from Eckard Specht. the diameter of the circle is the minimum distance between all pairs.

square lattice is optimal for 36 circles.

49 circles pack into a square smaller than 7x7: 6.974212543296849 .

50 circles require a square of side 7.005047458202193 , so given a 7x7 square, 49 is still max.

64 circles pack into a square of side 7.879104680207495 .

68 circles pack into a square of side 7.997430841597078 . this is the max under 8x8. straightforward to turn this into a puzzle. is the puzzle very easy to solve, just requiring hexagonal close packing?

what if the bounding square has rounded corners to exactly fit a circle in the corner? it probably does not matter.

similar questions can be asked about packing M*N circles into an M-by-N rectangle. when can M*N+1 circles fit into an MxN rectangle? when can M*N circles fit in a smaller rectangle that fits within an MxN rectangle (both dimensions less or equal)?

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