Solving a nontransitive dice problem I have a dice problem I would require some help with (especially problem b). Here goes the problem: 
Player 1 and Player 2 are playing a dice game, where both first select one die and then they throw the die. The one who gets a larger number from the throw wins. Player 1 chooses a die first and then Player 2 chooses one. The dice are the following:  A: 2,2,2,5,5,5
B: 3,3,3,3,3,6
C: 1,1,1,4,4,4 

So for example Player 1 could choose first die A, and then Player 2 could choose die B. 
a) What die should Player 1 choose so that he always has advantage (i.e. > 50 % chance of winning) no matter what die Player 2 chooses?
b) Player 2 is allowed to design a new die and replace it with die C. Then again Player 1 chooses a die first and then Player 2 makes the selection. What kind of a die should Player 2 design so that he would always have the winning advantage (i.e. > 50 % chance of winning), no matter what die Player 1 chooses?
How to approach this problem? Is there an analytical solution for b?
A: The answer to (a) is B. This can be easily manually checked.
Regarding (b); the reason B is the best die is that it "wins" both A and C, so player 1 will always want to choose it. What player 2 should do is replace C with a die which wins B, but which doesn't win A (else player 1 will choose it). So we have a non-transitive losing relation $\{(A,B), (B,C), (C,A)\}$.
Such a die can be 1 4 4 4 4 4.
