A few days ago Katie and I played the following game: one of us would say a number and the other one would have to make a rectangle out of that many squares (actually we used the board and chips from the game Othello). At first, everything was going well; we made nice rectangles for 8, 6, 9, and 12. For 12, after Katie made a 2×6, I pointed out to her that one can also make a 3×4 rectangle, which she was also familiar with. But then I gave Katie the number 7, and no matter how hard she tried, she couldn’t make a rectangle.
Katie: You tricked me! You make one.
I made a skinny 1×7 rectangle, but somehow Katie wasn’t satisfied. I had to confess that no other rectangle could be made out of 7 and told her that such numbers were called prime.
Katie: Ok, now I’m going to give you one.
And after a bit of thought she said…11! I think that she was basically trying to avoid the evens, and she also knew that a 3×3 rectangle contained 9 chips, so that one was no good. Hence, the fact that 11 turned out to be prime was at least partially a coincidence. Still, I was very pleasantly surprised.
We then went on to verify that 11 was indeed prime, as were 2,3,5, and 13. In the process, we made more rectangles for some composite numbers.
Musings of a Mathematical Mom 
Well, if Katie had been trying to avoid multiples of 2 and 3 (ignoring 1), then up through 24, she would only have gotten primes, so it’s not really so much of a coincidence that she got one. Composites always have “small” prime factors! I remember being very surprised as a kid when my father showed me that you could check that 47 was prime just by checking divisibility by 2, 3, and 5.
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You’re right, it’s not exactly a coincidence, but she didn’t know that it was sufficient to check 2 and 3, and in that sense she got lucky.
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