Where is the Missing Dollar?

Puzzles to fool your brain!

Posted Dec 01, 2009

I have often been asked what makes a puzzle interesting. Is it difficulty level or complexity? I don't think so, although difficulty and complexity do make a puzzle more challenging. The classic puzzles are actually rather uncomplicated in design. Their appeal lies in hiding some simple pattern or concealing an unexpected "twist" or "trap." These, I have found, create the greatest amount of frustration in solvers, but also generate the greatest amount of interest.

Especially frustrating (and interesting) is the type of puzzle that presents information that seems to defy logic. It is said that the Sophists-a group of traveling teachers who became famous throughout Greece toward the end of the fifth century BCE-made up these kinds of puzzles to expose the susceptibility of human logical thinking to deception and cleverness.

A classic in this genre is the so-called "missing dollar" puzzle, which never fails to confound solvers who come across it for the first time. As far as I can tell, the inventor is unknown. The earliest version I have been able to locate is the one published by R. M. Abraham in his 1933 book, Diversions and Pastimes, making him the most likely congener of the puzzle. I stand to be corrected. Solver beware! The trap is in the way the information is presented.

Three women decide to go on a holiday to Florida. They share a room at a hotel which is charging 1920s rates as part of a promotional strategy. The women are charged only $10 each, or $30 in all. After going through his guest list, the manager discovers that he has made a mistake and has actually overcharged the three. The room costs only $25. So, he gives a bellhop $5 to return to them. The sneaky bellhop knows that he cannot divide $5 into three equal amounts. Thus, he pockets $2 for himself and returns only $1 to each woman.
Now, here's the conundrum. Each woman paid $10 originally and got back $1. So, in fact, each woman paid $9 for the room. The three of them together thus paid $9 times 3, or $27 in total. If we add this amount to the $2 that the bellhop dishonestly pocketed, we get a total of $29. Yet the women paid out $30 originally! Where is the missing dollar?

Here is another puzzle of this type, which for some solvers is even more frustrating.

Yesterday, the first customer in a bookstore gave the salesclerk a $10 bill for a $3 book. Having no change, the clerk took the $10 bill across the street to a clothing store to get it broken down into ten $1 bills. The clerk then gave the customer the book worth $3 and seven $1 bills as change.
An hour later the clothing-store salesclerk brought back the $10 bill demanding her money back, claiming that the bill was counterfeit. To avoid quarreling, the bookstore salesclerk decided to give her ten $1 bills, taking back the counterfeit. What's the gist of the transactions that took place? The bookstore salesclerk was out $3 (= cost of the book), plus the $10 bills he gave to the clothing-store salesclerk. Altogether he lost $13. But only $10 were used in the transactions! What happened?

Since antiquity, we have prided ourselves on being a logical species. According to legend it was the Greek philosopher Parmenides who invented logic while he sat on a cliff contemplating the world. The French philosopher René Descartes refused to accept any belief or concept, including his own existence, unless he could "prove" it to be logically true. But puzzles such as these warn us that logic is not a foolproof tool of truth. It can be turned on its head to deceive the brain. As long as it sounds logical, we accept it as true. But that is not the case in real life, don't you agree? And then what is "logic" after all? Tweedledee put it satirically in Carroll's Through the Looking-Glass: "if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic."


First Puzzle: The trap in this puzzle is to be found in the way in which the numerical facts are laid out. The manager kept $25 of the $30 he was given. The women got back $3 ($1 each). So, far this adds up to $25 + $3 = $28. The remaining $2 dollars was, of course, pocketed by the bellhop. There is no missing dollar.

Second Puzzle: Like the previous puzzle, the deception here is in the layout of the numerical information. First, the bookstore salesclerk received nothing for the $3 book, since the counterfeit $10 bill was worth nothing. From the outset, he was out $3. That $3 went to the customer.
Now, consider what happened in the other transaction-the one between the bookstore and clothing-store salesclerks. The former received ten genuine $1 bills from his clothing-store colleague. So, at first it was the clothing-store salesclerk who was out $10. When the bookstore salesclerk got back to his store, he gave $7 of the ten good bills to the customer, and put the remaining good $3 in his pocket. The outcome of this transaction was as follows: the bookstore salesclerk was out another $7, while the customer gained $7. Altogether, the customer gained $10-a $3 book and $7 in good bills. That ends the bookstore salesclerk's transaction with the customer.

Now, the bookstore salesclerk was out the $3 for the book, not the $7 that he gave back as change to the customer-that came out of the pocket of the clothing-store salesclerk. When the clothing-store salesclerk asked for her $10 back, the bookstore salesclerk still had the $3 in his pocket left over from the $10 she had given him previously-the other $7 went to the customer. So, he gave her back her $3 and made up the $7 difference from his own pocket. In total, therefore, the bookstore salesclerk was out the $3 book and the $7 from his pocket-$10 in total.