I have two piles of marbles. Every round I select the pile with the least marbles, and double it, taking the necessary marbles from the other pile. If I start with 1 and 24, will either pile at some point be empty? What if it's 1 and 25, or 1 and 23?
You will only be able to reach 0 on the largest pile when the number of marbles is (2^k-1) Consider for instance that case when you have 1 and 15 marbles You will get this sequence of events: 1 15 2 14 4 12 8 8 0 16 This happens because by the time the smallest pile becomes larger than the other one you have subtracted 1+2+4+...2^k = 2^(k+1)-1. If you start off with more than 2^(k+1)-1 you can see there is no way you get one pile to zero.
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