Three men and a hotel

Posted by Steven Bryant On February - 7 - 2006

The Puzzle

Three men go to a hotel looking for a room for the evening. While all had reservations, the hotel only had one room available. The desk clerk offers to put them into the same room for the night and promised to resolve the problem in the morning. Knowing that the closest hotel with availability was over an hour away, the three men agreed to the arrangement.

The desk clerk informs them that the cost was 30 dollars. Each man takes 10 dollars from his wallet and gives it to the clerk. He hands them the key and they retire to their room for the evening.

Later that night, the manager returns from dinner, reviews the recent check-ins, and discovers the situation with the three men. She feels guilty about the mix-up and asks a bellhop to give them 5 dollars as a refund for the mistake. On the way to the room, the bellhop realizes that he cannot give each man an equal amount. Just then, an idea strikes him. He will keep 2 dollars for himself and give each man a dollar. He reasons that they will be happy because they get a refund. He knocks on the door and all three men answer. He hands them each a single dollar, and as he expected, they are happy.

Instead of paying 10 dollars, each man has now paid 9 dollars. 9 times 3 is 27, plus 2 is 29. Where is the extra dollar?

The Answer

Clearly there isn’t a missing dollar. In this case, we have to realize that once the manager decided to pay the men a refund of 5 dollars, the number 30 no longer had any meaning. The hotel was getting 25 dollars. So, the correct equation is written as 9 times 3 men is 27, minus the 2 “employee delivery fee” is 25.

What Does It Mean?

This is an example of derivation where every mathematical statement, taken individually, is correct. The problem is only revealed when the equations are taken in the context of one another. In this case, the last statement, 9*3+2 is 29 is mathematically correct. However, it does not belong in the story since it does not mathematically follow from the other statements.

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