Saturday, January 12, 2013

Answers to yesterday's brainteasers

Yesterday, for your amusement and edification, Walt presented The Missing Dollar , a brainteaser devised by R.M. Abraham included in his book Diversions and Pastimes (1933). Today, as promised, I'll give you the solution. But first, in case you missed it, here's the puzzle.

Three women check into a motel. The women are charged $10 each for their rooms, or $30 in total. (Hey, this was in 1933, remember!) Later, the manager discovers that he has accidentally overcharged the three vacationers. Their rooms cost only $25 in total, so he gives a bellhop $5 to return to them.

The sneaky bellhop knows that he cannot divide $5 into three equal amounts, so he pockets $2 for himself and returns only $1 to each of the women.

Here's the conundrum. Each woman paid $10 originally and got back $1. So, in fact, each woman paid $9 for her room. Therefore, the three of them together paid $27. 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 other dollar?

And now the answer, from an article by Marcel Danesi in the University of Toronto Magazine (Spring 2007).

The trap in this puzzle is not to be found in any single word, but in the way the numerical facts are laid out. The manager kept $25 of the $30 he was given. The three women got back $1 each. Together, that amounts to $28 ($25 + $3). The remaining $2 was pocketed by the bellhop.

Another way of looking at it... The guests paid $27 in total: $25 to the manager and $2 to the dishonest bellhop. There is no missing dollar.

Did you get it? Yes? Let's see if you managed to hit the perfecta. Here's the second conundrum.

A customer in a bookstore gives a sales clerk a $10 bill for a $3 book. (Still 1933, I guess.) The clerk, having no change, takes the $10 bill across the street to the record store (definitely 1933!) To get it broken into $1 bills. (Canadian readers can substitute loonies for the singletons.) The bookstore clerk returns and gives the customer the $3 book and seven $1 bills in change.

An hour later the record store clerk brings back the $10 bill, claiming that it is counterfeit. To avoid quarrelling, the bookstore clerk gives the record store clerk ten $1 bills and takes back the $10 note. How much has the bookstore clerk lost?

May we have the envelope, please...

Many people guess $13, but the answer is $10. The bookstore clerk received nothing for the $3 book, since the counterfeit bill was worth nothing. So, for the outset, he was out $3.

The bookstore clerk then received 10 genuine $1 bills from his colleague at the record store. He gave 7 of the 10 good bills to the customer, and kept the remaining 3 bills in his pocket. When the record store clerk asked for the $10 back, the bookstore clerk still had the 3 good bills. He gave them to the record store clerk and made up the $7 difference from the cash register. So he was out an additional $7. In total, the bookstore clerk was out $10.

If you went two for two, congratulations. If you didn't get either of them, welcome to the Society for the Numerically Challenged. Walt is a charter member.

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