Brain Teaser Tuesday: Where Is the Missing Dollar?

This is another brain teaser problem I came across when I was probably around 12 years old. I’ll admit it stumped me back then, even after I knew the answer, and it still does so today.

Here’s the riddle:

Three guests check into a hotel room. The manager says the bill is $30, so each guest pays $10. Later the manager realizes the bill should only have been $25. To rectify this, he gives the bellhop $5 as five one-dollar bills to return to the guests.

On the way to the guests’ room to refund the money, the bellhop realizes that he cannot equally divide the five one-dollar bills among the three guests. As the guests aren’t aware of the total of the revised bill, the bellhop decides to just give each guest $1 back and keep $2 as a tip for himself, and proceeds to do so.

As each guest got $1 back, each guest only paid $9, bringing the total paid to $27. The bellhop kept $2, which when added to the $27, comes to $29. So if the guests originally handed over $30, what happened to the remaining $1?

I’ll let you think about it for a while.

If you’re ready for some answers/explanations, feel free to read a detailed solution and history of the problem, on Wikipedia.

As I noted at the start, I’ve been aware of the various explanations for close to 50 years, but I still fall for the logical reasoning as to why the initial thought process was incorrect.

There are many variants of the puzzle. Professor David Singmaster’s Chronology of Recreational Mathematics refers to these type of mathematical problems as misdirection puzzles, and offers as another example, from 1880, the following:

Barthel sees two boxes at a jeweler’s, priced at 100 and 200. He buys the cheaper one and takes it home, where he decides he really prefers the other. He returns to the jeweler and gives him the box back and says that the jeweler already has 100 from him, which together with the returned box, makes 200, which is the cost of the other box. The jeweler accepts this and gives Barthel the other box and Barthel goes on his way. Is this correct?

Of course it’s not correct, but it’s a pretty slick move by Barthel, and the jeweler falls for it.

To me, the takeaway is to charge the right amount for the hotel room right from the start, and this way you don’t have to deal with the dishonest clerk.

And as for the jeweler, maybe you need to keep an accountant around to handle the returns…

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