Why are there 62,208 puzzles?
This last Friday my niece turned 9, and someone gave her this “Genius Square” as a gift.
Note the label on the left hand side of the box:
62,208 puzzles, and there’s always a solution. Hmm, such an odd number. Why 62,208? It look like it’s close to a power of 2, and indeed 216 = 65,536, but this isn’t really helpful. I suppose at this point I should explain the game and how we ended up finding out exactly how the authors got to 62,208.
You start by rolling a special set of dice that come with the game (they’re at the top of the image below). Then you take the little wooden pegs and place them in the location specified by the dice.
Then the puzzle, the object of the game, is to take those pieces along the sides and make them fit while respecting the peg locations.
Alright, now that we’ve gotten the preliminaries out of the way let’s talk about how we get 62,208 puzzles.
Attempt #1: Well there’s 7 dice, and dice are usually 6 sided, so is it 67? Nope, that’s 279,936, way too many.
Attempt #2: Well, the game board is 6x6, and there’s 7 pegs, so maybe this is like an n choose k situation? Nope, 36 choose 7 is 8,347,680. We’re getting colder.
Attempt #3: Let’s take a closer look at those dice. I wrote down all the faces of the separate dies and here’s what I found:
Immediately die VI should jump out at you. It only contains 2 options despite having six sides! Is that why we get such a strange number of puzzles? Maybe it’s 66 · 2? Nope, that’s 93,312, but we’re getting close!
As it turns out die V also has two repeats, F2 and A5, (and by the way all squares are covered, we checked), so that gives us 65 · 4 · 2 = 62,208. □
As to the math behind how they ensured that each puzzle was solvable, and whether or not that has anything to do with the selection of squares on the dice, I have no idea! My niece and I did figure out that the puzzle solutions are not unique, but that’s about as far as we got.