Analogias/Memes

A riddle about jigsaw puzzles.

As my four-year-old gets more and more into jigsaw puzzles, my role as father has narrowed to a single, satisfying, Zen-like task:

Sorting edge pieces from middle pieces.

A riddle about jigsaw puzzles.

Not long ago, as my daughter tackled a 7×7 puzzle, I noticed that the two species of pieces — middles over here, edges over there — looked pretty similar in size. A quick calculation verified it: they were similar in size. The puzzle was 7×7 = 49 pieces, and the interior was 5×5 = 25 pieces, leaving 24 pieces for the edges. (You can also calculate the number of edges directly as 4 edges times 7 pieces per edge, minus the 4 corner pieces that have been double-counted. Again, 24.)

That’s a mere one-piece difference. The puzzle was just about half edge.

This led me to a puzzle about puzzles: Are there any rectangular jigsaws with precisely the same number of edge pieces and interior pieces? And if so, what are the largest and smallest such puzzles?

I found that question satisfying. But I wanted more. And so I began thinking about 3D puzzles — or, as I preferred to imagine them, modular space stations. Picture cube-shaped rooms assembled into rectangular prisms, drifting through space.

Now, instead of edge and interior pieces, we’re counting modules with windows, and modules without. The question: Are there any space stations with precisely the same number of windowed and windowless modules? What are the largest and smallest such space stations?

I hope you find the riddles as pleasurable as I do. And watch out for spoilers, which I’ll allow in the comments below. I’ll chime in with some comments about why I love these puzzles — for now I’ll just say it relates to my book chapter titled “The Square-Cube Fables.”

P.S. You’ll notice that I have stopped at 3D, although one could certainly extend the puzzle to 4D and beyond. At that point, the mind ought to turn to from specific solutions to questions of scaling. As the dimension grows, what scaling behavior do we see for the number of solutions, and for the N-dimensional measures of the largest and smallest solutions? Beats me!