A safe contained five slips of paper, each of which had a dollar amount written on it. Two of the amounts were negative. On the outside of the safe was a big sign with the total dollar amount within (which was positive).
A ring of robbers break into the safe, each takes one slip of paper, and they return to their hideout, and sit down in a circle. They all know the total amount of money, but only now does each one now look at his own slip of paper to see his share. Of course, the ones with negative amounts are quite unhappy, because they've lost money on this heist!
One of the robbers who has a negative amount on his paper decides to alter the results. He knows that the total amount of the heist cannot be changed, but he decides to borrow the slips of paper of his two neighbors, subtract from each slip the amount that he owes, and replace his own negative amount by a positive amount. Thus, if his slip originally read "-$10,000", he would subtract $10,000 from both of his neighbor's amounts, and rewrite his slip to read "+$10,000", and then replace his neighbor's papers. Of course, he's happy now.
We said there was a second robber with the same problem. Surely, he's still in the hole (maybe even further in the hole if he sat next to the first guy). So he does the same thing.
Now it's possible, even likely, that after these two steps there may be one or more robbers who are in the hole. If so, let us allow them to carry out the same operation. In fact, as long as any robber has a negative amount of money, they will be allowed to do this.
The question is: must this process eventually stop? must it go on forever? does it sometimes go on forever and sometimes not?
I give up, show me the solution.
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