Then I told her *her* age was a perfect number, and we defined that for her (perfect numbers). She asked me whether there were any odd perfect numbers, and I said if you didn't consider 1 one, then I didn't know. I'm glad to know I wasn't missing any obvious deductions there. ;-) She said that 1 and 0 are special numbers and often exceptions. She asked if primes could be perfect numbers and we thought aloud about that until she agreed that they couldn't.
I said I could think of one other perfect number that was fairly small, that she could count to within a minute. She said that meant a number up to 20, so I had to revise my statement slightly. She pounced on that with glee. "That's a clue!" and I had to admit that the number I was thinking of was higher than 20. We then played an arithmetic 20 questions as she narrowed in on what number it might be. (Is it over 30? Is it prime? (No, we'd already figured out it wouldn't be prime. Oh, right.) Is it odd?) Once she'd essentially figured out what number it must be, she wouldn't say it out loud; first she had to confirm it for herself by finding and adding up the factors, so she went around muttering numbers while she worked that out.
She needed help with division at several points, but she'd never outright ask the answer to the question she was working on, but would simplify it as best she could and then ask. So it wasn't "is 28 divisible by 4?" Instead she clearly found 28-(2*4) first and then asked "is 20 'even in 4s'?" Which is how the six-year-olds say it these days, I guess.
There was a rough spot where she thought 13 was a multiple of 3 and 5, and I had to keep track of all the factors and prompt her to add them up when she'd found enough of them to be interesting, but it was really an impressive bit of work. I wish I'd had someone to play numbers aloud with like that when I was a kid.
And then it was a couple minutes past her bedtime, so I read her a chapter of a Charlie Bone book while she used rmd for a jungle gym, and then it was time for lights out. Hee.