Nim - Ahead of the Game

[@khoda81]

Every now and then I find some random underrated channel on KZbin and you are, by faaar, the most underrated channel I have ever seen. Absolutely amazing videos❤

[@mostly_mental]

Glad you like it. Thanks for watching!

[@jmanmac10]

You have a fascinating and intriguing channel and everything is presented very well. New sub earned and thank you very much!

[@mostly_mental]

I'm glad you like it. Thanks for watching!

[@ArslanRozyjumayev]

Pure Gold!

[@mostly_mental]

Glad you liked it. Thanks for watching!

[@urcool3116]

Is there a way to calculate the Nim sum in the head? cus its kinda impossible and slow as soon as there are bigger numbers

[@mostly_mental]

Not that I know of. When I work with game theory, I'll usually just plug numbers into a computer. Most opponents would call that cheating, but Aurora is very patient with me.

[@charnelion]

1. Pairs of the same numbers cancel out: 6 ⊕ 10 ⊕ 10 = 6 3 ⊕ 3 ⊕ 3 = 3 2. When all numbers are powers of 2 (except for the lowest number, which doesn't have to be), it works like addition: 2 ⊕ 16 = 18 1 ⊕ 2 ⊕ 4 = 7 3 ⊕ 4 ⊕ 8 = 15 3. Any positive integer can be written as a nim sum of the greatest power of 2 it contains and the remainder: 5 = 4 ⊕ 1 9 = 8 ⊕ 1 11 = 8 ⊕ 3 15 = 8 ⊕ 7 21 = 16 ⊕ 5 81 = 64 ⊕ 17 We know that the same terms cancel out, so: 11 ⊕ 9 = 8 ⊕ 3 ⊕ 8 ⊕ 1 = 3 ⊕ 1 = 2 You can take any two numbers (or four, six, etc.) which share the same greatest power of 2 and just subtract it from them: 3 ⊕ 4 ⊕ 5 = 3 ⊕ 0 ⊕ 1 = 2 4 ⊕ 5 ⊕ 6 = 4 ⊕ 1 ⊕ 2 = 7 8 ⊕ 9 ⊕ 10 ⊕ 11 = 0 ⊕ 1 ⊕ 2 ⊕ 3 = 0 4. If the larger of two numbers is one less than a power of two, then we have subtraction. 3 ⊕ 0 = 3 3 ⊕ 1 = 2 3 ⊕ 2 = 1 3 ⊕ 3 = 0 7 ⊕ 0 = 7 7 ⊕ 1 = 6 7 ⊕ 2 = 5 7 ⊕ 3 = 4 7 ⊕ 4 = 3 7 ⊕ 5 = 2 7 ⊕ 6 = 1 7 ⊕ 7 = 0 15 ⊕ 0 = 15 15 ⊕ 1 = 14 15 ⊕ 2 = 13 15 ⊕ 3 = 12 ...and so on.