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Using the Ponting Square Packing, squares of size 1-49 can be packed in a 7x7 array so that the 25 interior squares are completely surrounded.

ponting 49

Another way to look at the above squares is with the following grid:

Ponting grid 7

Is there some similar method that allows cubes of size 1 to 125 to be packed so that 27 interior cubes are completely surrounded? If so, would the central section of cubes have a cross section similar to the following?

central 125 section?

If such a 3D packing exists and is extendible, what does the shape look like for a lot of cubes? As an example, here are squares of size 1 to 2025.

ponting 2025

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  • $\begingroup$ Explain something about how the package is formed. $\endgroup$ – Piquito Jul 9 at 19:41
  • $\begingroup$ In your figure, just $9$ and $16$ are "completely surrounded" ? What does mean for you "completely surrounded"?, please. $\endgroup$ – Piquito Jul 9 at 20:02
  • $\begingroup$ In the first figure, square 19 is surrounded by 26, 29, 33, 30. Square 30 is surrounded by 12, 27, 15, 26, 19, 33, 16, 34. $\endgroup$ – Ed Pegg Jul 9 at 20:06

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