by
Erik Oosterwal
The quick answer is that both men were wearing a white hat.
There are four possible ways that the hats could be distributed among the
two men: 1) Both are wearing white hats, 2) the first is wearing white
and the second wearing black, 3) the first is wearing black and the second
wearing white, 4) or both wearing black. We know that the 4th option
isn't valid because they were told at least one hat was white.
If the first man had seen a black hat on the other man he would have known
instantly that his hat was white. If the second man had seen a black hat
on the first, he would have instantly known his hat was white.
Since some time passed with neither claiming that the other was wearing a
black hat, they suddenly concluded that they were wearing a white hat.
The quick solution is that Bart exclamed he was wearing the opposite color
than Andrew was wearing.
There are eight ways that black and white hats can be distributed among three
men:
Andrew |
Bart |
Clifford |
|
| 1. | W |
W |
W |
| 2. | W |
W |
B |
| 3. | W |
B |
W |
| 4. | W |
B |
B |
| 5. | B |
W |
W |
| 6. | B |
W |
B |
| 7. | B |
B |
W |
| 8. | B |
B |
B |
We were told that at least one hat was white and at least one hat was black,
so options 1 and 8 can not be valid. That leaves options 2 through
7. If the hats were arranged as in position 2, Clifford would have
immediately known he was wearing a black hat. Similarly, if the hats
had been arranged as in position 7, Clifford would have known he was wearing
a white hat. Were were told that some time passed before anyone said
anything so we can conclude that Clifford did not see two of the same color
hat in front of him. This leaves only positions 3, 4, 5, and 6 as valid
solutions. In each of these we can see that Bart is wearing the opposite
color from Andrew. Bart concluded that since Clifford could not tell
for sure what color his own hat was that his (Bart's) must be different from
Andrew's.