We used to play a game back when I was at school. We would start with three coins placed widely apart on a table. The first player would give one of the coins a shove, so that it would pass between the other two. Then the next player would try to do the same, starting from the current position, but using one of the other coins. This would continue until one player would inevitably fail to get the coin between the other two.

I’m now imagining an idealised version, where the coins are single points, and on each turn the coin passes through the exact midpoint of the positions of the other two coins and continues a further 50%. So to begin with, the coins are at (0,0) (3,9) and (9,3) respectively. The midpoint of the initial positions of coins B and C is clearly at (6,6), so coin A will travel from (0,0) to (6,6) and then continue another 50% to come to rest at (9,9). Next it is the turn of coin B. Coin B must travel to the current midpoint of coins A and C and extend another 50% distance to find its new position. The coins take turns in the order A, B, C, A, B, C etc

The diagram below shows the first two moves of each coin.

After an infinite number of moves, where will each of the three coins be?