The total amount of money possessed by all 100 players is $\displaystyle 100X$ and this is constant throughout the games. Suppose the 63rd player has $\displaystyle Y$ amount of money at the start. After 62 games, he will have $\displaystyle 2^{62}Y$ money. The total amount of money of the other players is $\displaystyle 100X-2^{62}Y.$ Thus, after the 63rd game, the 63rd player gives this amount of money in total to the other players, and he is left with $\displaystyle 2^{62}Y-(100X-2^{62}Y)=2^{63}Y-100X$ in his pocket. The remaining 37 games are then played, after which the 63rd player has $\displaystyle 2^{37}(2^{63}Y-100X)=2^{100}Y-2^{39}25X.$

Hence

$\displaystyle 2^{100}Y-2^{39}25X\ =\ X$

$\displaystyle \implies\ \fbox{$Y\ =\ \dfrac{(1+2^{39}25)X}{2^{100}}$}$