Before we begin, let's look at a simulation of this problem where the total is $45,000, but three robbers got negative amounts. We will list the amounts, in thousands, that each robber has, and presume that on each step the robber with the most negative amount initiates an exchange.
| +15 | -10 | - 5 | +50 | - 5 |
| -10 | +20 | -10 | ||
| + 5 | +10 | -15 | +50 | - 5 |
| -15 | +30 | -15 | ||
| + 5 | - 5 | +15 | +35 | - 5 |
| - 5 | +10 | - 5 | ||
| 0 | + 5 | +10 | +35 | - 5 |
| - 5 | - 5 | +10 | ||
| - 5 | + 5 | +10 | +30 | + 5 |
| +10 | - 5 | - 5 | ||
| + 5 | 0 | +10 | +30 | 0 |
Thus, after 5 exchanges, the swapping stops, because no one is actually negative. But was this just an accident? Will the system always settle down? Does it matter how many robbers, or how much they took?
Back to the The Ring of Robbers Puzzle.