I heard this years ago and have never come up with a satisfactory answer:
Three men take a room at a motel. It costs them $30 and each contributes $10. The motel manager realises he has over charged them and gives the bell boy 5 dollar bills to return to the men. The bell boy, realising that 3 doesn’t go evenly into 5, simplifies the mathematics by returning $1 to each of the men and trousers the remaining $2 for himself.
The men originally paid (3x$10) $30. Now that they have been given a dollar each bach, they paid (3 x 9) $27. With the bell boy’s $2 this makes $29. Where is the missing dollar?
The “mystery” is the result of obfuscatory wording of the problem – the $2 can’t be added to the $27, because they are already present in the math! The $27 is reached by doing the sum 30-5+2 – which is the sum of money leaving the guests’ wallets.
Already answered well, but just in case this phrasing makes it any clearer:
It doesn’t make any sense to try adding the bellboy’s $2 to the $27 that they paid, because it’s still a part of the amount that the men paid. If you take $2 for bellboy away from the $27 paid, you end up with the $25 room charge to the hotel.
If you add the $3 refunded to the $27 paid, then you get the $30 originally paid out. Nothing is ‘missing’ if you look at the numbers in the way that matches what happened.
“I don’t know. But listen, I know of this great motel downtown where you pay $30, and the bellhop gets $5 to give back to you, but he only gives you $3 and keeps $2. And $27 x $2 = 54 dollars squared! I didn’t even know they made square dollars! But watch out for the seedy one across town. Because there, the honest bellhop divides your money out evenly. $27 / $5 = 5.4. 5.4 what, I have no idea.”
Hopefully they realize that if they did the only remaining operation, 27-2, they’d have no problem figuring out who’s got the 25 bucks.
I like to illustrate the answer by changing the numbers. They pay $30. Then the manager realizes that today is free motel day and gives the $30 to the bellboy. The bellboy gives back nothing and keeps it all for himself. According to the puzzles logic, the men paid $30 and the bellboy has $30, adding those gives you $60. Where did the extra $30 come from?
Usually they can work out from there what’s wrong and apply it to the original problem.
At each stage there’s always 30 dollars. The trick is picking some arbitrary numbers from the final step and pretending they’re supposed to add up to 30 when really they don’t have any relation to the original amount.
The last sentence above should read “With the bell boy’s $2 this makes $25”. The $2 must be subtracted, not added (just as the 3 x $1 returned was subtracted) in order to reconcile to the amount the hotel received.