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A standard die shows a random number of pips between 1 and 6. Can we do the job with fewer pips? I can tell you how (though not why anyone would want to!). See my Mathematical Enchantments essay "Industrious Dice", with a link to a video of my G4G15 talk. https://mathenchant.wordpress.com/2024/10/17/industrious-dice/
@JimPropp I was inspired recently by your industrious dice to come up with this d6 design, where pips can be either at the vertices or centers of faces. Opposite sides (which are given the same color) sum to 7 as in standard d6's.
@two_star Lovely!
@JimPropp But not the most efficient! It has 18 pips, which is barely an improvement over the standard 21.