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.
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!
Love this riddle, Ben! Thanks for posing it.
I’ve added a link to your puzzle at the bottom of my “Puzzle Pastimes” post https://karendcampe.wordpress.com/2020/05/20/puzzle-pastimes/
Spoiler warning for solutions to the 2D, 3D, and a start to the general problem:
Taking the problem as: Interior=Exterior (I=E), we can add interior to both sides to get:
2I=I+E=C, or the whole cuboid is twice the interior. The whole shape is lwh…, and the interior is (l-2)(w-2)(h-2)…. Dividing both sides by the interior, we get (l/(l-2))(w/(w-2))…=2, or (1+2/(l-2))(1+2/(w-2))…=2. This naturally extends to arbitrarily many dimensions (including 1d, for a length 4 “puzzle” with 2 “edges”).
In 2D, taking L<W, the only choices for L/(L-2) are 5/3 or 6/4 (they must be between sqrt(2) and 2). This gives two puzzle sizes, 5×12 and 6×8, so a 60 piece or 48 piece puzzle respectively.
In 3D, L<WL and W/(W-2)>sqrt(14/9)), and neither of those have a height that works. If L=8, we must have W=8,9, or 10, for similar reasons. It turns out 8x8x18, 8x9x14, and 8x10x12 all work, for 1152, 1008, and 960 pieces respectively. For L=7, you get 7x7x100, 7x8x30, 7x9x20, and 7x10x16 (4900, 1680, 1260, and 1120 pieces). For L=6, you get 6x9x56, 6x10x32, 6x11x24, 6x12x20, and 6x14x16 (3024, 1920, 1584, 1440, and 1344 respectively). Finally, for L=5, you get 5x13x132, 5x14x72, 5x15x52, 5x16x42, 5x17x36, 5x18x32, 5x20x27, and 5x22x24 (8580, 5040, 3900, 3360, 3060, 2880, 2700, and 2640 pieces respectively).
In total, you have 20 possible solutions for 3D, with the largest being 5x13x132=8580 modules, and the smallest being 8x10x12=960 modules. I am surprised by how many possibilities actually work. I would imagine the combinations would grow exponentially (maybe even faster) for higher dimensions. The solution 2D+2x2D+4x…x4D works for any number of dimensions (4, 6×8, 8x10x12). I had thought it would continue to give the smallest examples, but 11x12x13x15 slightly edges it out in 4D (25740 rather than 26880 for 10x12x14x16). It does seem like roughly factorial growth in the n dimensional volume of the shapes.
I brute-forced it with some Python code; nested for loops, so embarrassing. Anyway, 2D and 3D were no problem, but my laptop got whiny and incalcitrant about 4D, reluctantly gave me 307 solutions in a 500-to-a-side tesseract and then tapped.
In a 7×7 world, a task so neat, Counting the pieces, a puzzle feat.
An update! I hadn’t been able to find these jigsaw sizes anywhere in the OEIS, but it turns out that there’s a sequence counting the number of solutions in each dimension: https://oeis.org/A263207 The number of solutions does appear to be super-exponential in dimension, as Thomas suggested.
This puzzle is quite difficult