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Three Men Rent a Hotel Room - A Semantic Fallacy
By Cynthia Kirkeby
Aug 6, 2008, 10:27

An employee of mine has posed this question to me and I can not come up with an answer.

    Three men rent a hotel room. Each pays $10 for a total of $30 spent on the room. The next day the hotel owner tells the three men that they over paid for the room as it only costs $25. The three men tell the owner to give them each a dollar back and he can keep two dollars.

    If you do the math, each man paid $9 a piece for the room for a total of $27. The owner kept $2 which brings the total to $29.

    The question is where did the other dollar go?

This question is driving me nuts. Can you help?


This actually isn’t as much a math problem, as a logic and semantics problem. It took me a few minutes to spot the logical fallacy in this problem. I knew it was there, but it was difficult to spot nonetheless.

This is a semantic fallacy. The premise is false, which you can see when you work the equation backwards. The men do pay a total of $9.00 each (total of $27.00) for the room, but only after you add in the amount they tipped the owner. They paid $25 dollars for the room and they gave the owner $2 as a tip. ($25 + 2 = $27.00). At this point the only thing to add back in to get to the original $30.00 is the $3 the men received as a refund. ($27 + $3 = $30)

Or to simply it further:
There is a misdirect in the problem when it says that when you do the math each man paid $9 each for the room, plus $2 tip. They don't. They pay a combined price of $25.00 plus the $2 tip. The problem sets up a set of facts:
  • $30.00 was originally given to the hotel owner
  • Correct price of the room = $25
  • Each man receives $1 dollar refund = $3
  • The owner receives a $2 tip
  • $25 for room + $2 tip for the owner = $27
  • $30 - $27 = $3 which is how much the men receive back as a refund.

Those are the facts. Once you separate those out, you can see the misdirect in the puzzle. There isn't a dollar discrepancy as listed, all the money is accounted for.

Wording things in misleading ways is what allows swindlers to make a living. By wording it the way they have in the problem, you actually add in the $2.00 to the owner twice, and you don’t add in the $3.00 rebate to the men at all, which leaves you short $1.00.

Cynthia Kirkeby
ASK ClassBrain

Tony D has written it up another way... As you will see, we're essentially saying the same thing. The assumption in the original premise of $9 per man is incorrect. I worked the problem backwards and he works the problem forwards: "I read this riddle and realized, like you, that this seemingly mathematical impossibility is more a matter of semantics and deception than it is a mystery. The solution is so easy if you focus on the facts. The confusion begins with the explanation that the men…”paid $9.00 a piece…”. The $9 is not a fact; they didn’t pay $9 a piece; they paid $10 a piece. The math should start from the $10 not the $9. The solution thereafter is simple: Therefore: $10 x 3 = $30 $30 - $2 (amt owner kept) = $28 $28 - $3 (refund to renters) = $25 cost for the room" Once again, the math is simple if you ignore the misleading information set up in the puzzle. Keep your wits about you and watch out for semantic tricks.

Another note: Roxie, an accountant of 30 years wrote and said this about the problem: "The misdirect is NOT in saying the men paid $27.00 because that IS what they paid. The misdirect is in adding the owner's $2.00 to their $27. There are no semantics involved here, just poor math."

The misdirect in the problem which prompts you to add the $2 tip to the $27.00 is semantics. It is the logical fallacy that makes this problem perplexing at first glance.

Roxie continues by saying, "But like I said before......The final amount paid by the 3 men was $27.00 and the final amount received was $27.00 ($25.00 to the owner and $2.00 to the bellboy). Simple (and correct) answer.... I haven't been an accountant for 30 years for nothing, and believe me 2 + 2 will ALWAYS equal 4 (unless of course you work for the government......but that's a whole other issue)."

We would like to thank Roxie for giving us yet another look at the fundamentals of this problem.

Here is another version of the problem. The solution is the same as above. Watch out for the misdirect.

3 Men Go Into A Motel. The Man Behind The Desk Said The Room Is $30, So Each Man Paid $10 And Went To The Room.

A While Later The Man Behind The Desk Realized The Room Was Only $25, So He Sent The Bellboy To The 3 Guys' Room With $5.

On The Way, The Bellboy Couldn't Figure Out How To Split $5 Evenly Between 3 Men, So He Gave Each Man A $1 And Kept The Other $2 For Himself.

This Meant That The 3 Men Each Paid $9 For The Room, Which Is A Total Of $27, Add The $2 That The Bellboy Kept = $29.

Where Is The Other Dollar?

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