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There are various studies for packing sequential squares of size $1$ to $n$. We can try to find the smallest square they will pack into, as in tightly packed squares. We can find the smallest square they will fit into, A005842. We can find an optimal stacking for them.

Adam Ponting found an efficient packing for squares of size $1$ to $(2 n+1)^2$, as seen in his article square packing. I expanded that for Ponting Square Packing. In the below, an offset is used on his method to pack squares of size 32 to 200. A value of 31 can be subtracted from all sizes for a packing of squares 1 to 169.

Ponting packing 32-200

So, what is a square packing? For the purposes of this article, I'll define it as follows:

  1. Up to 4 squares can have 2 fully exposed edges. All other squares must have 2 or more edges fully covered and a third at least partially covered.
  2. No holes are allowed in the packing.

Ponting mentions only finding packings for an odd square number of sequential squares. I took a look at order-4 and sizes representable in a 4x4 matrix and found 8 solutions. The second one has squares with sizes {{11, 8, 15,6},{7,12,5,16},{9,4,13,2},{3,10,1,14}}.

even square packings

I haven't been able to find solutions with a 6x6 or higher even matrix. Can anyone find those, or other square packings of sequential squares?

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