Game of Coins

Kuku and Pipi decide to play a game. They arrange 50 coins in a line on the table, with various nominations. Then alternating, each player takes on his turn one of the two coins at the end of the line and keeps it. Kuku and Pipi continue doing this, until after the 50th move all coins are taken. Prove that whoever starts first can always collect coins with at least as much value as his opponent.


Solution

Let's assume Kuku starts first. In the beginning he calculates the total value of the coins placed on odd positions in the line and compares it with the total value of the coins placed on even positions in the line. If the former have bigger total value, then on every turn he takes the end coin which was placed on odd position initially. If the latter have bigger value, then on every turn he takes the end coin which was placed on even position initially. It is easy to see that he can always do this, because after each of Pipi's turns there will be one "odd" coin and one "even" coin at the ends of the line.