Now it was Alice's turn. “If you want to play a game, I've got a good one for you. Just as you wanted, first one to score a hundred wins. You roll a pair of dice. If you roll doubles, I win $2$ points. If the two dice have a difference of one, I win $1$ point. If the difference is $2\text{,}$ then it's a tie. If the difference is $3\text{,}$ you win one point; if the difference is $4\text{,}$ you win two points; and if the difference is $5\text{,}$ you win three points.” Xing interrupted to say “In other words, if the difference is $d\text{,}$ then Dave wins $d-2$ points.” Alice continues “Right! And there are three ways Dave can win, with one of them being the biggest prize of all. Also, rolling doubles is rare, so this has to be a good game for Dave.”