Helen was not convinced that the two times table contains the same number of numbers as the seven times table. [A conversation she initiated herself going to bed, out of nowhere.] She understood the idea of using a bijection to show two collections have the same cardinality without counting them - I modelled it (conceptually, not physically) with smarties, buttons and a lot of string - and she could see how 2n <-> 7n works, but (not surprisingly) it just seemed wrong to her. Just wait till she finds out the rationals are countable!

Like most kids, she wants to know about bigger and bigger numbers, but I told her it might be a few years before I could show her a bigger infinity (and that she had to learn place-value fractions - "decimals" - first). And for large finite numbers, she needs to understand ordinary exponentiation before I can explain the Knuth up-arrow to her. (Though if I ask her to draw a red-blue edge-coloured K₅ that doesn't contain a monochromatic K₃ she'll probably just start trying it.)

Helen doesn't actually know her seven times tables yet. She's done a tiny bit of playing Times Table Rock Stars, but seems keener on learning structural patterns. So she knows her squares and does 6x7 by subtracting 7 from 49, and is learning (N-1)(N+1) = N²-1 - so 7x9 = 64-1.

**Update:** I explained the Infinite Pigeonhole Principle to Helen — that if an infinite number of pigeons are put into a finite number of pigeonholes, at least one hole must end up with an infinite number of pigeons — and she pointed out that it was also possible to put an infinite number of pigeons into each pigeonhole.

She has also pointed out (out of the blue again) that there is no nearest number to *ya* (which is 64 in the base-4 "Banana" number system we learned from Paul Lockhart's *Arithmetic*), because one has *ya bat* (64+1/4), *ya lat* (64+1/16), *ya yat* (64+1/64), etc. (using an extension to Lockhart's system we invented ourselves).