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.

Each guest got $1 back, so each one only paid $9. That means the total paid was $27. The bellhop kept $2, which when added to the $27, comes to $29. If the guests originally handed over $30, what happened to the remaining $1?

The Missing Dollar Riddle has been in circulation since at least the 1930s. It has been known to stump the most brilliant of minds, rob its contemplators of sleep, and make even the calmest of people scream in frustration.

On its face, the riddle seems to reflect the impossibility that all of us have experienced from time to time: money just disappears from our pockets, and we have no way to account for it. A closer look shows us that something else is going on that has less to do with math or economics and more to do with the sleight of hand and misdirection used by magicians and grifters.

The riddle hinges on redirecting the solver’s attention to math that is properly done with improperly-grouped numbers. In the third paragraph of the riddle, numbers are added together under the assumption that they should total $30. In reality, there is no reason the numbers should add up to this amount.

Adding the $9 that each of the guests paid and the $2 tip to the bellhop does, indeed, total $29. What it fails to consider is that this simply represents the total of a smaller amount that changed hands. In order to account for all of the money that changed hands, it is necessary to include the $25 retained by the clerk.

In other words, consider the amount retained by the clerk ($25). Add to that the $1 returned to each of the guests ($3). Add on the “tip” retained by the bellhop ($2): $25+$3+$2=$30. Everything is accounted for.

Another way to explain this is to understand that the $27 already includes the bellhop’s tip. To add the $2 to the $27 would be to double-count it. The cost of the rooms for the three guests, including the “tip” totals $27. Each of the guests got $1 returned. When added to the $27 revised cost of the room (including “tip” to the bellhop), the total is $30.

One reason the riddle works is because of the relatively small amount of the bellhop’s “tip.” If you are still having difficulty understanding or explaining the riddle, try it again, but use larger numbers. The fallacy becomes a lot more evident:

Three people check into a hotel room. The clerk says the bill is $30, so each guest pays $10. Later the clerk realizes the bill should only be $10. To rectify this, he gives the bellhop $20 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn’t know the total of the revised bill, the bellhop decides to just give each guest $6 and keep $2 as a tip for himself. Each guest got $6 back: so now each guest only paid $4; bringing the total paid to $12. The bellhop has $2. And $12 + $2 = $14 so, if the guests originally handed over $30, what happened to the remaining $16?

If all of this math is giving you a headache, take a look at the way Bud Abbott and Lou Costello made use of the same principle to generate laughter instead of frustration.

Categories: Brainteasers and Puzzles, Humor, Mathematics, Money

That’s the best brain teaser I ever read

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Thanks! I’m glad you enjoyed it.

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Well that was fun! Thanks for the explanation – otherwise I would still be trying to figure it out …oh, and can you loan me 50 cents?

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How about if I loan you forty, and then I’ll just owe you ten?

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I accept!

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