The Missing Dollar (math/logic puzzles!)

Previous posts about Chief Theresa Spence -- the icon of the Idle No More "movement" -- have mentioned the damning auditor's report into mismanagement of the millions of dollars doled out by the Canadian government (read Canadian taxpayers) to her band council. Under the supervision of her boyfriend -- a bankrupt -- taxpayers' money just vanished, paid to persons unknown without proper documentation or, indeed, sometimes without any documentation.

The story reminded Walt of an article that appeared in the University of Toronto Magazine about six years ago. Marcel Danesi wrote about "contrary-to-expectation" puzzles, designed to show how susceptible human logic is to the power of deceptive language.

The use of numbers and language to confuse the reader was known to the ancient Greeks, and persists to this very day in such things as misleading advertising. Case in point: TV infomercials that promise you a great deal on, say, padded bras, and then tell you they'll send you a second set of the same, absolutely free. Of course you pay extra for the shipping and handling of the bigger package, but hey, it's free, right?

This works because many people have trouble understanding mathematical logic if a set of numbers is presented in a deceptive manner. A classic brainteaser (Danesi writes) was described by R.M. Abraham in Diversions and Pastimes (1933). Here is The Missing Dollar.

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?

Figured it out already? Here's another one.

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?

Answers tomorrow.