5 patches

Problem

A tyrant, having among his prisoners 3 scholars whom he esteemed for their intelligence, proposed the following proof to them:

"Here are 5 color patches, 2 black and 3 white. Each of you will have one attached to his forehead. Then you will be led to a room where you will be disposed so that you can see one another. The one who first finds out what color patch he is wearing will go free immediately. "

- adapted from: E. von Glasersfeld (2009) Partial Memories. Sketches from an Improbable Life. Exeter: Imprint Academic, p.91

Table 1: Five patches puzzle

ANALYSIS

As a prisoner, what could I see and what would the other two prisoners have seen? For answering this, on a wax tablet I would have sketched a table of all cases (see Table 1). The table shows that there are 7 cases with 21 configurations, but only 3 different types:

     * type A: I see 2 black patches; this type has 3 instances, hence 3/21 = 14% of probability, that I would see it.
     * type B: I see 1 black and 1 white patch; this type has 12 instances, hence 12/21 = 57% of probability, that I would see it.
     * type C: I see 2 white patches; thiy type has 6 instances, hence 6/21 = 28% of probability

PREPARATION

Given these 3 types and probabilities, how would I prepare myself for the challenge?

Type A (instances: 1, 5, 9):
Since there were only 2 black patches in the set, my patch can only be white
IF I see two black patches, THEN my patch is white
INFERENCE A => quick, independent inference.
Unfortunately the probability of A is low (14%); moreover the tyrant would probably avoid this configuration, because it is too easy for the prisoner who receives the white patch.

Type B (12 instances):
* IF my patch would be black (instances 2, 3, 4 & 6, 7, 8), two of the three patches would be black, THEN the prisoner with white would see two black patches and by inference A could immediately tell, that he has white.
* IF the prisoner with white does not react immediately AND he payed attention THEN I can be sure that I have not black (instances 11, 12, 13 & 15, 16, 17)
INFERENCE B => delayed inference, depending on another prisoner's reaction and state

Type C (6 instances):
* IF my patch would be black (instances 10, 14, 18), THEN the other two prisoners would see a black and a white patch and would need to do the delayed inference of type B (in group II)
* IF the other two prisoners don't seem to do the delayed inference of type B THEN I could infer that my patch is white (instances 19, 20, 21).
INFERENCE C => delayed inference, depending on the two other prisoner's reaction.

Five patches

Problem

Table 1: Five patches puzzle

ANALYSIS

PREPARATION

SOLUTION: always say immediately WHITE!