## Ask Professor Puzzler

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Steve from Cincinnati writes: "I believe this puzzle answer is 1089.

11 X 11 = 4

22 X 22 = 16

33 X 33 = ???

"The first two equations are wrong, but they trick us into looking for a pattern rather than apply the mathematical rules...many people assume an addition function exists between each of the two pairs of integers. Not so. Therefore, thirty three multiplied by thirty three is 1089."

Hi Steve! Of course, you are 100% correct. 33 x 33 = 1089, no matter how many incorrect equations you put before it. Now that I've said that, if I stop there, I've got a really short blog post, and no one wants that - so let's fix their notation in order to turn it into a more interesting puzzle. We'll treat it as a single variable function:

Find x if:

F*(*11) = 4

F(22) = 16

F(33) = x

It *still* isn't a very good puzzle, though, because there's not enough information to come up with a single solution. In that respect, it's very similar to the "Squares Problem" I wrote up last year. Here are just three possible answers to the puzzle.

**Solution #1**

We add the digits of the number in the function's domain, and then square it. (1 + 1)^{2} = 4; (2 + 2)^{2} = 16; (3 + 3)^{2} = **36**.

**Solution #2**

We square the number, and then add the digits: 11^{2} = 121 and 1 + 2 + 1 = 4; 22^{2} = 484 and 4 + 8 + 4 = 16; 33^{2} = 1089 and 1 + 0 + 8 + 9 = **18**.

**Solution #3**

We subtract 8 from the number, and then add the quotient the number and eleven: 11 - 8 + 11/11 = 4; 22 - 8 + 22/11 = 16; 33 - 8 + 33/11 = **28**.

I'm sure there are other solutions; those are just the ones that popped into my head on a quick glance. Can you come up with others? Send them my way and I'll add them to this post.