Discuss what students found, and why they think it’s true. If students did the multiplication correctly, they’ll have noticed that every digit works. But why?

There are a few good explanations I can think of, and students may have more. Here are two examples:

*Explanation 1:* (1 + 1 + 1) x 37 = 3 x 37 = 111.

If replace the ones by twos, that would double the answer, so you’d get 222; and so on for threes, fours, etc. up to nines.

*Explanation 2:* (7 + 7 + 7) x 37 = (7 x 3) x 37 = 7 x ( 3 x 37) = 7 x (111) = 777, and likewise for other digits.

Students may or may not have found these solutions. If not, you don’t have to give it to them. They’ll have gotten the multiplication practice they need, and the mystery can be something to continue thinking about later.