Home > Riddles > The envious dice

The envious dice

Chance and uncertaintyLevel 4 · Advanced · ●●●●○

Four special dice:

A: 4, 4, 4, 4, 0, 0
B: 3, 3, 3, 3, 3, 3
C: 6, 6, 2, 2, 2, 2
D: 5, 5, 5, 1, 1, 1

You choose a die first. The dealer then chooses another die.
Whoever rolls the highest number wins.
The dealer states: “no matter which one you choose, I can choose a die that favors me with probability $2/3$”.
How can it be true? What is your strategy?

Solve this riddle

Hints

Show hints

Dealer strategy (choose second): If you choose A, he takes D.
Key comparisons: P(A>B)=2/3 (A scores 4 in 4 of 6 cases).
Yes, the dealer can always respond with a 2/3 probability of winning.

Solution

Show full solution

Back to problem
Answer: Yes, the dealer can always respond with probability of victory $2/3$.
Key comparisons:

$P(A>B)=2/3$ (A gets 4 in 4 of 6 cases).
$P(B>C)=2/3$ (C rolls 2 in 4 of 6 cases).
$P(C>D)=2/3$.
$P(D>A)=2/3$.

A non-transitive cycle is formed:

$$ D>A>B>C>D. $$

Dealer strategy (choose second):

if you choose A, he takes D;
if you choose B, take A;
if you choose C, take B;
if you choose D, take C.

In all cases, $2/3$ is favored with exact probability.

Related riddles

Keep practicing

If you enjoyed this one, try more pure-logic riddles, explore this theme, browse the full archive, or read the riddle-solving guide.

← Previous: Nim (3,4,5) in misère mode · Next: The tournament without a referee →