Even and odd

Abi likes to annoy grown-ups by playing "the opposite game", where everything she says is the opposite of what she means. Once, a few months ago, the possibility somehow came up of playing the "opposite opposite" game; we figured out that in this game, everything you said was true. What about the "opposite opposite opposite" game? How about 4 opposites, or 5, or 6? Soon the games were referred to by their numbers: "I'm playing the seven game". Then Abi had a revelation: "It alternates -- opposite, non-opposite, opposite, non-opposite." And then another: "All the numbers that end in 1,3,5,7,9 work one way, and the ones that end in 2,4,6,8,0 work another way." We had already told her about even and odd numbers at some point, but I don't remember if she figured out this connection herself or if I pointed it out to her. In any case, she got very adept at telling what kind of game a given number corresponded to. She also picked her favorite two games to play: when she wanted to play the opposite game, she said she was playing 69 (=34+35, the sum of her parents' ages). When she wanted to play the truth game, she said she was playing 74 (= 34+35+5). From all of this, she definitely got the idea that even and odd matter (though she doesn't quite remember the names yet, in English or Russian: she just says "numbers that you can divide in half").

Last night, Josh and Abi were counting by 3's. This morning, Abi told me she had figured out something very interesting: 3,6,9 is like 30,60,90. "In what sense?" I asked. She had a little trouble expressing it, but finally explained that 30,60,90 are the first "tens" (des'atki -- the conversation was in Russian) that occur when you count by 3's, and they're the same as the first numbers that occur. I asked what the next few "tens" would be, and she told me: 120, 150. Then I asked if the same thing would work with counting by 7's, but she said that was too hard: let's try counting by 4's. At first she conjectured that the first "ten" would be 40, but I told her to try, and she figured out it was actually 20. At that point, she seemed quite sure that if you counted by 5's, the first "ten" would be 50. When she realized it wasn't, she said she had thought it had to do with evens and odds, but apparently not.

(At that point she was sick of the conversation: she likes doing math a little at a time, but tires of it quickly, and we never push.)