Radio Astronomy Systems Research Group

Bill is wearing a Black hat.

To find the solution, you must go through all possibilities.

Tom's options are B/W/W, ?/B/B, ?/B/W, or ?/W/B: (Black, White or ? (unknown))

  B   W   W
 Tom Bob Bill -- Tom sees white hats and knows he has a black hat

  ?   B   B       ?   B   W       ?   W   B
 Tom Bob Bill or Tom Bob Bill or Tom Bob Bill -- Tom sees B/B, B/W, or W/B so he doesn't know his hat color.

Since Tom doesn't know, Bob's options are ?/B/B ?/B/W or ?/W/B.

  ?   B   W
 Tom Bob Bill -- Bob sees Bill has a white hat, and knows he has a black hat.

  ?   ?   B
 Tom Bob Bill -- Bob sees Bill has a black hat, so he doesn't know his own hat color

Since Tom and Bob don't know, Bill only has one option: ?/?/B

  ?   ?   B
 Tom Bob Bill -- Bill knows that Tom and Bob don't know, so they must see a black hat on Bill's head.