Home > Riddles > The envious dice
These says seem comparable in a normal way, but they hide a circular relationship: each one can be defeated by another. The advantage is not in choosing the “best” die, but in choosing later.
There are four special dice: - A: 4, 4, 4, 4, 0, 0
| Answer: there is no one die that is better than all. The second player can always choose a die that beats the one chosen by the first with probability $2/3$. The key comparisons are: | Comparison | Why it wins | Probability |
|---|---|---|---|
| A beats B | A wins if he rolls 4; B always rolls 3 | $4/6=2/3$ | |
| B beats C | B wins if C rolls 2 | $4/6=2/3$ | |
| C beats D | C wins if he rolls 6, or if he rolls 2 and D rolls 1 | $2/6+(4/6)(3/6)=2/3$ | |
| D beats A | D wins if he rolls 5, or if he rolls 1 and A rolls 0 | $3/6+(3/6)(2/6)=2/3$ | This is how a cycle is formed: $$ |
A>B>C>D>A.
$$ Therefore, the second player always responds with the die that beats the one chosen by the first: - against A, choose D;
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