Pairwise Coin-Tossing Game
Problem 605
Consider an -player game played in consecutive pairs: Round takes place between players and , round takes place between players and , and so on and so forth, all the way up to round , which takes place between players and . Then round takes place between players and as the entire cycle starts again.
In other words, during round , player faces off against player .
During each round, a fair coin is tossed to decide which of the two players wins that round. If any given player wins both rounds and , then that player wins the entire game.
Let be the probability that player wins in an -player game, in the form of a reduced fraction. For example, and .
Let be the product of the reduced numerator and denominator of . For example, and .
Find the last digits of .