## Sunday, January 31, 2021

### [ejdnedtu] 4900-piece pyramid-square puzzle

4900 square tiles: arrange the tiles in a 70-by-70 square to form a picture, similar to a regular jigsaw puzzle, except pieces don't interlock.

flip all the tiles over and rearrange the tiles into a 3D square pyramid: bottom layer 24-by-24, next layer 23-by-23, and so forth, up to one tile at the apex.  if done correctly, the pyramid viewed from above forms a different picture.

tiles have image portions printed on both sides.  to make things easier (if desired), texture or 3D shape can help distinguish which side is to be used for which structure.  tiles could also have a shape, like Lego, to help keep the layers aligned when building the pyramid.

pyramid has many tiles partially or completely obscured.  obscured parts can be left blank (making things easier) or have misleading false images printed on them.  or, there could be many possible ways to assemble the pyramid all having the same final shape but different pictures when viewed from above.

picture on the square side: Times Square.  picture(s) on the pyramid side: Giza pyramid complex.

the mathematical magic is 70^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2 + 13^2 + 14^2 + 15^2 + 16^2 + 17^2 + 18^2 + 19^2 + 20^2 + 21^2 + 22^2 + 23^2 + 24^2 which has something to do with the Leech lattice and maybe the Monster group.  this is the only (nontrivial) equation of this form: 4900 pieces is the only number of pieces a puzzle like this can have.  cannonball problem.

if one or both sides of the puzzle are truncated square pyramids (unfinished pyramids), then there are more possibilities.

future work: investigate (truncated) rectangular pyramids.  a m-by-n layer gets a (m-1)-by-(n-1) layer above it and so forth, up to 1 by (1+abs(m-n)).

triangular or hexagonal tiles.

instead of tiles, consider cubes.  up to 6 different puzzles (with different final structures) could use the same collection of cubes.  the total number of cubes required will probably be enormous.

back to 4900 square tiles: what is a good brick-shaped case to pack the tiles for storage?  14*14*25, 10*10*49, 7*7*100.