Two players, a gambler and dealer, play continuous blackjack. The gambler goes first and generates a uniformly random number in (0,1). At any time, they can choose to stop or play again. If they play again, they generate another independent uniform random number in (0,1). If the sum of the generated numbers exceeds 1, the gambler busts and the dealer wins. Otherwise, if the gambler stops at some value 0≤a≤1, then the dealer begins generating independent uniform random numbers in (0,1). The dealer keeps adding the values up until he either obtains a sum in the interval (a,1) (in which case the dealer wins) or obtains a sum larger than 1 (in which case the gambler wins).
The strategy of the gambler is going to be to stop when the sum is at least α and hit (generate another random number) if the sum is below α. The α that maximizes the probability of the gambler winning solves eα=x+yαe+α for integers x and y. e here is Euler's constant. Find x2+y2.
