Unlike other fake proofs on Everything, this doesn't take the standard tack of exploiting algebraic or arithmetic errors, and thus is much more effective at truly confounding people's smug sense of truth:

Three businessmen stop at a hotel and decide to evenly split the cost of the only vacancy, a $30 double. Upon arriving in their room, the bellhop discovers that it is only a single, and should cost only $25. He returns to the front desk and retrieves five dollar bills. On his way back to the room, the unscrupulous bellhop realizes that the five dollars won't go around evenly, so he pockets the two extra and hands over the three remaining dollars to the three businessmen.

So, the three businessmen each paid $9 (the $10 minus the $1 refund) for the room, a total of $27. The bellhop has the remaining extra $2 in his pocket. That adds up to ... $29. WTF??? Where'd the dollar go?


This paradox is based on highly misleading logic. The final summary is not an real representation of the amount of money in circulation at a point in time. We may say that the businessmen now have $3, the bellhop has $2, and the receptionist has $25. This adds up neatly to $30.
I really like this one. Kudos to blaaf for a good puzzler, but for the longest time, I simply could not account for the final sentence:

So, the three businessmen each payed(sp) $9 (the $10 minus the $1 refund) for the room, a total of $27. The bellhop has the remaining extra $2 in his pocket. That adds up to $29.

While his solution gave an alternate way of looking at things, it did not give a reason as to why that last sentence was misleading. After a long period of deliberation, I came to the following conclusion:

In the beginning, the businessman paid $30. All the money in circulation (a total of $30) originally came from the businessmen and was held in the hotel's possession. After the $3 rebate, $25 was in the hotel's possession, $2 went to the bellhop, and the other $3 was held in the businessmen's possession. Therefore, the phrase "Each business man paid $9 for a total of $27" is true ($25 + $2).

The point of error in the aforementioned sentence is adding the bellhop's money to the money paid by the businessmen. Where did the bellhop's money come from? It came from the businessmen's originally paid money! The $2 didn't appear from nowhere, now did it? The $2 is part of, and not in addition to, the $27 that the businessmen paid. Add the $3 (rebated money that the businessmen now possess) and now you come to the $30 originally in circulation.

Whew!

wh00t's explanation, while entirely correct, didn't point out the logical fallacy quite clearly enough for a friend of mine (who shall remain nameless, Ian), so here it goes, for the slow ones:

The original premise:
The businessmen payed $9 a piece, for a total of $27, plus the bellhop's $2 makes $29, so where's the extra dollar?

The corrected premise:
The businessmen payed $9 a piece, for a total of $27, *minus* the bellhop's $2 makes $25, the price of the room.

Hope that clears up any remaining confusion.

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