Critical Thinking Problem on Mark's Color

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Here is the description you provided of the problem faced by the three men:

A mark is placed on each man's forehead. The three are told that each has either a black mark or a white mark, and each is to raise his hand if he sees a black mark on the forehead of either of the other two. The first one to tell what color he has and how he arrived at his answer will get the job. Each man raised his hand, and after a few seconds one man comes up with the answer. What color was his mark, and how did he figure it out?

We are not told how many men get black marks, so there are four possibilities: 0 men, 1 man, 2 men, or 3 men have black marks. It might be helpful to visualize these situations as below. (The subscripts for numbering the men are only for convenience in referring to them; order doesn't actually matter—so in Situation II for example, it wouldn't matter which man had the only black mark)

Situation I:
W1

W2 W3

Situation II:
B1

W2 W3

Situation III:
B1

W2 B3

Situation IV:
B1

B2 B3

We are told that all three men raise their hands.
• So the men know they are not in Situation I where no one would raise his hand.
• And they know they are not in Situation II where man 1 would not raise his hand.
• In Situation III all of the men would raise their hands: man 1 sees a black mark on 3; man 2 sees black marks on 1 and 3; and man 3 sees a black mark on 1.
• And of course in Situation IV all ...