This week’s puzzle is a modification of a brain teaser that has been around for many years—the hotel problem. In the original version of the mathematical paradox, three men pay ten dollars each for a thirty dollar hotel room. Later, they receive a five dollar refund. Because they cannot easily split the five dollars, they each keep one dollar and give the bellboy a two dollar tip. Thus, each man’s share of the room is nine dollars. Nine dollars multiplied by three is twenty-seven dollars. When the two dollar tip is added, the total is twenty-nine dollars. The conundrum comes in determining what happened to the missing dollar. In the modification presented here, the paradox is the same, but the setting is slightly different. The challenge remains for students to find the missing dollar.

Each student should be encouraged to persevere in discovering a solution and not to share the answer with their classmates once it is discovered.

**Solution Hint: **Think about what is being added to reach the total of twenty-nine dollars.

Three men agree to share a cab from the airport into town. When they arrive, the meter reads $25. Each man gives the driver a $10 bill. She hands them five $1 bills as change. Each man takes one of the $1 bills. They give the driver the remaining two $1 bills as a tip.

Each man has now spent nine dollars, and the driver has two dollars, bringing the total to $29 (3 x 9 = 27, 27 + 2 = 29). What happened to the other dollar?

Your challenge is to explain this paradox.

**Solution**

Click the arrow below to view the solution.

To explain this paradox, you need only to add up the amounts. The fare was $25, the tip was two dollars, and each man kept one dollar. This adds up to $30. The error in reasoning is caused by the sentence that says, “Each man has now spent nine dollars and the driver has two dollars for a total of $29.” Actually the first part of this sentence is correct—each man does spend nine dollars, since the total cost of the cab ride is $27 (the $25 fare and the $2 tip). The error comes in the second part of the sentence. What needs to be added to this 27 dollars is the three dollars that the men keep, not the two dollar tip—the tip was accounted for in the cost of the ride.

Of course it was duhhhhh!

[…] in at least five valid ways depending on the assumptions you make. Last week’s puzzle was “Cab Conundrums,” which is another good and hopefully enjoyable math problem. Try one or both of these problems […]