I do love the Beast Academy books. My 3rd Graders are working on multiplication, and the Beast Academy books have these crossword puzzles. The kids love them:

(Why do the kids love them? Oh, I don’t know. If interest = “this is new” times “I can do this” then I guess this has enough going on that it feels new. And the puzzle is self-checking, which probably validates that “I can do it” feeling. That’s all I’ve got.)

Every puzzle can both be solved and studied. I’ve made it a habit to encourage my students to ask questions about the puzzles we solve, and I usually do this by sharing a question or two that I have.

My question was, can you make this puzzle using only multiplication?

And my kids’ questions were:

- Can you make one of these crosswords only using subtraction?
- Are the blanks really necessary?
- Could the puzzle be smaller? Larger? Could it be 5 x 4? 8 x 2?
- Could it be shaped like a path?

The next class, I gave students some blank crosswords and asked them to see if they could fill in the blanks in a way that worked.

Which was interesting. But the kids wanted more, so I sat down to make more crosswords in the original style. The original puzzles always include mostly multiplication, and then one addition and one subtraction equation, always on the right and bottom sides.

So I set out to make a few puzzles in this style. I started filling in the boxes, and got stuck. Then I tried again — still not working. I started to get that familiar good/bad feeling that happens with math. It’s the feeling of “oh this is harder than I thought” but also “there might be something here!”

Over lunch, I interrupted two of my colleagues and recruited them into the problem. (I was happy to return the favor after they’ve done the same to me so many times.) We filled out the crossword with variables.

Using these variables, the puzzle is only possible if . My colleague pointed out that you might factor this a bit and then solve for d:

A few things about that equation:

- It means that the whole puzzle is determined by just three of those variables.
- d is a whole number, so (b + c) needs to go into a(b – c).

This is not a ton to work on, but suppose that the sum of b and c is chosen to be a prime number. It clearly won’t go into (b – c). So that means a will have to be a multiple of b + c.

That seems to work!

This leaves me with a bunch of questions, though. Does this characterize all the possible crossword puzzles? I feel like this finds one specific way of getting a crossword that works here, but is it really the only way? Also, I haven’t really thought about whether I could use any multiple of b + c. I think I can, just because it’s worked whenever I’ve tried it so far, but it would be better to understand why.

There’s a math textbook that I like that makes the case that there is significant mathematics that has been developed by teachers, just for the sake of having nice examples to give to students. I always like when that sort of thing happens, a nice mathematical surprise that appears sometimes when you remember to look for it.