Where Did The Other Dollar Go?

(This post first appeared on June 13, 2009. Today's update provides a solution to the proposed problem.)

I don’t remember when I first heard this riddle, but I recognized it right away as a sweetie. I’m offering a Quack of the Cold Duck itself as a prize for the first solution. If there isn’t a prize winner in a few days, I’ll fill in the blanks. Here’s the riddle:

Three buddies walk into a bar and order a pitcher of beer. After downing a glass or two, they decide to order a large pizza loaded with the works. The waitress takes their order and tells them that the pizza costs $30. They each toss a $10 bill on the table, and after pocketing the cash, the waitress heads for the kitchen.

Meanwhile, one of the boys uses the men’s room, and on the way back to his table runs into the waitress. She apologizes, telling him that she had the price wrong. Since the pizza only costs $25, she hands the fellow 5 singles back. Thanking her for being honest, he gives her $2, then pockets $1, and gives $1 to each of his two buddies back at the table.

So each buddy spent $9 counting the 3 $1 refunds. And the tip was $2. That makes $27 plus $2, or $29.

Where did the other dollar go?

June 18, 2009

Like a magician’s trick, this problem is all about misdirection. The charm of this misdirection is that the $1 difference is so small that solvers don't smell the rat in the problem itself, but rather doubt their own addition.

Each boy in fact spent $9. Each boy laid out $10, and each got a $1 refund. As a group, they spent $25 for the pizza and $2 for the tip. $9 times 3 = $27 = $25 + $2. There is no “missing” dollar.

The problem as phrased sets up a spurious equation.
By adding the tip to the $27 total cost, the problem adds the tip twice.

If this explanation doesn’t work for you, let’s try another one that avoids the original mental landmines entirely. Suppose you take a cab ride to the airport. The cabbie tells you that the fare is $25. You hand him 3 $10 bills, and ask for 3 singles as change; he can keep the $2 as a tip. No tricks here. This is the identical payment scheme found in the original problem, except that 3 people each get $1 returned in the original, while one person gets $3 returned here.

The misdirection in this problem is so powerful that my solution has been greeted more than once with a response like, “Well, your answer makes sense. But you still didn’t answer the original question.”

I guess I’ll just have to get smarter.

While waiting around for that to happen – I hope you’re all comfortably seated – let’s take a break and see how another mathematician struggles to correct his pals’ faulty long division.