The three prisoners and the red hats

1. The cry "RED!" It is not intuition: it is elimination of the hypothesis "I am blue" using the observed silence as logical information.
2. 2) Reference scenario with 2 red and 1 blue
3. The one who shouts "RED" uses reasoning by levels of knowledge (induction about how many reds there are).

Answer: The one who shouts "RED" uses reasoning by levels of knowledge (induction on how many reds there are).
1) Conceptual basis
Public rule: "at least one hat is red."
That generates a game of logical rounds:
-If someone saw two blue ones, they would instantly deduce that theirs is red.
2) Reference scenario with 2 red and 1 blue
If one prisoner were blue and the other two were red:

• every red would see (red, blue),
• after a first silence, each red would conclude that it cannot be blue and would deduce red.

That is to say: with exactly 2 reds, the deduction comes one round later.
3) Real case: 3 reds
Each prisoner sees two reds and reasons:

• "If I were blue, we would be in the case of 2 reds; then the other two would have already been able to deduce it after that first silence."
• "But that expected behavior did not occur."
• "Then I'm not blue."

Therefore he deduces that his hat is red.
4) Conclusion
The cry "RED!" It is not intuition: it is elimination of the hypothesis "I am blue" using the observed silence as logical information.