My family has a nice shut the box set at home. They play with the rules variation where the final score is the digits left uncovered, read in order -- ie, if 1, 5, and 9 are left uncovered, the score is 159 (some variants instead use the sum of the digits as the score), and you must roll two dice unless the sum of the uncovered digits is 6 or less, in which case you must roll only one die (some variants make this a free choice). In a two player game, both players put a dollar in the pot and then the first player attempts to shut the box; if he fails the second player starts from scratch and tries to shut the box; if they both fail, the winner is the player with the lower score. If the game ends by shutting the box, the winner gets the $2 in the pot and an extra dollar from the other player (double the stake); otherwise the winner just gets the pot.
An easy question: is this a fair game? If not, who is the sucker?
A hard question: what's the best "short" strategy for playing the game? (I'll define a "short" strategy as "capable of being described on a single side of a sheet of paper.)