Around the 37 minute mark Charlie mentions that the sum of odd numbers starting at 1 = (number of numbers) squared.I'm surprised Charlie wasn't aware of what the sum of integers is:https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AFsum(k) from 1 to n = n(n+1)/2Multiply by 2 and add 1 (in the sum) to get sum of odd numbers:sum(2k+1) = 2 * sum(k) + n => sum(2k+1) = n^2 from above.Gauss is said to have figured this out as an infant.Strange that this surprises Charlie so much.I didn't hear him mention it; but "casting out nines" is always interesting. Add the digits of 9 times anything to get 9 (or zero modulo 9 one would say).9 * 4 = 36; 3 + 6 = 99 * 727 = 6543; 6 + 5 + 4 + 3 = 18; 1 + 8 = 9.Charlie might want to take an abstract algebra course - or at least a number theory course. Or read the Bourbaki books (or similar). I bet he'd be surprised at how much these things come in handy. For instance: bar codes.
Oh jeez I spoke too soon. He went into "casting out nines" right after I hit "enter"!I'm serious about Charlie taking an abstract algebra course. Since this sort of thing obviously interests him, he'll find that many of these things he mentions are somewhat easily provable. But many are not. I found abstract algebra and Bourbaki stuff to be pretty interesting in my college days. There are actually alot of implementations of these relations in applied mathematics as well. Charlie would be inspired to no end with "conspiracy theory" type stuff from these relations!I'm not being sarcastic ... there are ALOT of interesting things based on number theory.
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