After lights out the other night Marcus started trying to tell me about something he had worked out with numbers. I asked him to hold it over to the next day. We forgot it about it until it was bedtime again, when I promised I would look at it with him on paper, the next day. At five to seven the next morning he was waving a pen and paper at my head. I rose reluctantly.
What he had worked out, while lying in bed, was this. The sequence of squares (starting with 0 x 0) goes 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 etc. Marcus started thinking about the gaps between the squares. These increase steadily by 2, so their sequence is 1, 3, 5, 7 , 9, 11 etc. I thought this was a really wonderful bit of pattern-finding. Having talked about it with a few people, I know that this is a very well known fact, but it had never occured to me.
It's probably time to look up that gifted children's organisation we heard about and get Marcus bumping brains with some other junior Einsteins.