As my four-year-old will get increasingly into jigsaw puzzles, my position as father has narrowed to a single, satisfying, Zen-like activity:

Sorting edge items from center items.

Not way back, as my daughter tackled a 7×7 puzzle, I seen that the 2 species of items — middles over right here, edges over there — regarded fairly related in measurement. A fast calculation verified it: they had been related in measurement. The puzzle was 7×7 = 49 items, and the inside was 5×5 = 25 items, leaving 24 items for the sides. (You can even calculate the variety of edges instantly as 4 edges occasions 7 items per edge, minus the 4 nook items which were double-counted. Once more, 24.)

That’s a mere one-piece distinction. The puzzle was nearly half edge.

This led me to a puzzle about puzzles: Are there any rectangular jigsaws with exactly the identical variety of edge items and inside items? And if that’s the case, what are the most important and smallest such puzzles?

I discovered that query satisfying. However I needed extra. And so I started occupied with 3D puzzles — or, as I most well-liked to think about them, modular area stations. Image cube-shaped rooms assembled into rectangular prisms, drifting via area.

Now, as an alternative of edge and inside items, we’re counting modules with home windows, and modules with out. The query: Are there any area stations with exactly the identical variety of windowed and windowless modules? What are the most important and smallest such area stations?

I hope you discover the riddles as pleasurable as I do. And be careful for spoilers, which I’ll permit within the feedback beneath. I’ll chime in with some feedback about why I like these puzzles — for now I’ll simply say it pertains to my e-book chapter titled “The Sq.-Dice Fables.”

P.S. You’ll discover that I’ve stopped at 3D, though one may definitely lengthen the puzzle to 4D and past. At that time, the thoughts ought to show to from particular options to questions of scaling. Because the dimension grows, what scaling habits will we see for the variety of options, and for the N-dimensional measures of the most important and smallest options? Beats me!