The game Flipper is played with 2 coins which may have any integer denomination from 1 to 99, inclusive. The coins are tossed, and the value of the toss is the sum of the denominations of those coins that show heads. The higher sum wins. In case of a tie, the game is replayed.
Two people play this game. One person has coins valued 1 and X, the other person has 5 and X. It is found that the game is fair, that is, the two people win in exact proportion to the total value of their coins.
What is the value of X?
I will generalize a bit and say that the two people have coins {A,X} vs {B,X} with X>B>A.
Person 1 can flip totals of 0, A, X, or A+X
Person 2 can flip totals of 0, B, X, or B+X
There are then 16 possible results of a match:
Person 1 wins with A>0, X>0, A+X>0, X>B, A+X>B, or A+X>X: 6 ways
Person 2 wins with 0<B, 0<X, 0<B+X, A<B, A<X, A<B+X, X<B+X, or X+A<X+B: 8 ways
The two people tie with 0=0 or X=X: 2 ways
Then by the given definition of a fair game we must have 6/8 = (A+X)/(B+X). Solving for X yields X=3B-4A.
This value of X needs to be greater than B, so we now have the condition 3B-4A>B, which simplifies to B>2A.
So if we have B>2A then we are guaranteed a solution X=3B-4A.
Fortunately the original problem conditions satisfy the inequality: 5>2*1 is true. Then 3*5-4*1=11 is the value of X that makes the original game fair.
Generalized Solution
