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July 4th, 2026 (Permalink)
Independence Day at the Logicians' Club
Those fun-loving logicians of the Logicians' Club (LC) celebrate Independence Day each year with a patriotic puzzle party, and what a wild party it is! This year's party was even wilder than usual since it celebrated the 250th anniversary of American independence. Here's how it went:
Every member wore a hat that was either red, white, or blue. The LC is, as I'm sure you know, a rather exclusive organization, so there were only six hats available: three red, two blue, and one white. The white hat was reserved for the president of the club, so all the regular members received either a red or blue hat. All of the club members were aware of these facts.
The president of the club was the first to arrive at the venue chosen for the party, and the first thing he did was make sure that the room had no mirrors or other shiny surfaces. When the first three members arrived, it was time to begin the festivities. Carrying the six hats in an opaque bag, the president approached each member from behind and placed a hat on the member's head in such a way that the member could not see its color. However, each member was able to see the color of the other two members' hats.
After the three members had each been issued a hat, the president asked one―Ms. X―what color her hat was.
"I don't know," Ms. X said, and the other two logicians heard her answer. The president turned to the second member, Mrs. Y, and asked the same question.
"I don't know," Mrs. Y said, and again the other two logicians heard her answer. The president turned to the final member, Professor Z, and asked the same question.
Prof. Z was just about to reply in the same way as the previous two logicians when he suddenly stopped and correctly gave the color of his hat.
What was the color of Prof. Z's hat? Are you LC material? If you can solve this problem, you are!
LC members will figure out the color of their hats from the information available if that's possible.
Members of the LC never lie, unless it is part of a game. Lying is not part of this game.
Red
Explanation: The important clue to this puzzle is that neither Ms. X nor Mrs. Y knew the color of her hat. The only way that Ms. X could know the color of her hat is if both Mrs. Y and Prof. Z were wearing blue hats, in which case X would realize that her own hat had to be red, since there were only two blue hats. So, when it comes Mrs. Y's turn, she knows that one or both of her and Prof. Z has a red hat. If Prof. Z's hat were blue, which Mrs. Y would see, then she would know her own hat must be red. Since she didn't know her own hat color, Prof. Z's hat must be red.
Previous Independence Day puzzles:
- A Patriotic Puzzle Race, 2004
- Independence Day Parade Puzzle, 2012
- A Fireworks Puzzle, 2013
- An Independence Day Puzzle at the Logicians' Club, 2019
- An Independence Day Patriotic Shoestring Puzzle, 2020
- A Second Meeting of the New Logicians' Club, 2021
- The Fourth of July Picnic, 2025