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MCIII Number Theory 2.6
Hello,
The solutions in the back of the book basically say the pattern is 2m for N=2^n+m. I test many numbers and of course they work, but I don't understand why.
Thanks,
Tina Jin
You can start doing the operations until you have $2^n$ cards left in the stack. How many steps have it been? Now that we have $2^n$ card, the bottom card will be the remaining one at the end. So what is the current bottom card?
I think I understand now. After m moves, the top 2m cards are now either discarded or below the (2^n+m)^th card. Only the even ones are below the (2^n+m)^th card, and the last one of those even cards (also the last card) would be 2m. The last card in a pile of 2^n cards will be the last card remaining since in a stack of cards with a power of 2 number of cards, it will be a skip the first, keep the second, skip the third, keep the fourth, ... skip the (2^n-1)^th, keep the (2^n)^th then repeat this process and only the (2^n)th card will always be kept.
Thanks!
Tina Jin
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