Medium

Logic

Alvin Wan

Nov. 26, 2016

Hats Off to The Enemy

Four logicians have been captured by The Enemy and will only be set free if one of the four logicians can win the game. The four logicians are placed in a line, so that the fourth can see the other three, the third can see the first two, the second can see the first, and the first cannot see anyone. The logicians are told that there are 3 white hats, 4 black hats total. The four logicians close their eyes, and The Enemy selects some subset of these hats, placing one hat on each of the logicians' heads. The logicians are instructed to state the color of their hat if they know it. The logicians open their eyes, and after a period of time, the first logician calls out the color of his hat. What color was it? How did he know?

Solution

Let us denote the 4 logicians as $L_1, L_2, L_3, L_4$.

What color was it?

$L_1$ has a black hat.

How did he know?

If $L_4$ sees three white hats, $L_4$ definitely has a black hat and would've shouted out. $L_4$ does not shout, so there is at least one black hat among $L_1, L_2, L_3$.

Knowing that $L_4$ is silent, $L_3$ knows there is at least one black hat among $L_1, L_2, L_3$. Thus, if $L_3$ sees two white hats, $L_3$ definitely has a black hat and would've shouted out. $L_3$ does not shout, so there is at least one black hat among $L_1, L_2$.

Knowing that $L_3$ is silent, $L_2$ knows there is at least one black hat among $L_1, L_2$. Thus, if $L_2$ sees one white hat, $L_2$ definitely has a black hat and would've shouted out. $L_2$ does not shout, so $L_1$ must have a black hat. Back to all Riddles