A and B each have two unbiased four-faced dice, the four faces being numbered 1, 2, 3, 4. Without looking, B tries to guess the sum x of the numbers on the bottom faces of A 's two dice after they have been thrown onto a table. If the guess is correct B receives x 2 euros, but if not he loses x euros.
Determine B 's expected gain per throw of A 's dice when he adopts each of the following strategies:
1. he selects x at random in the range 2 ≤ x ≤ 8;
2. he throws his own two dice and guesses x to be whatever they indicate;
3. he takes your advice and always chooses the same value for x . Which number would you advise?