Where Did the Dollar Go?

One of the visitors to Esmerel challenged our mathemagician with a riddle:
"Have you ever heard about the three men that were out on the town? After having consumed more drink than they should, they decided to wear it off by staying overnight in a hotel. The desk clerk charged them $30.00 for the room. Shortly afterwards the desk clerk realized that he had overcharged the three men by $5.00. He calls the Bell Boy over, gives him the $5.00 and explains to him that he had overcharged the three men and asked him to go up and give them the $5.00. On his way up, the Bell Boy thinks they will be happy to get a refund so why don't I give them each $1.00. They will be happy and I will have picked up $2.00. But when he does that and gives each man $1.00, that means they only paid $9.00 each for the room, which is a total of $27.00; plus the $2.00 the Bell Boy pocketed is a total of $29.00. But they gave the desk clerk $30.00! Where did the extra $1.00 go??"

Answer

Pacing back and forth in the snow, and tugging at his beard, the mathemagician replied:

Well. let's look at the initial situation. Each man pays $10 for the hotel room, the hotel gets $30, and the bellboy has nothing. Now, when everything has settled, each man has paid $9 for the hotel room (a total of $27), the hotel got $25, and the bellboy got $2. So what went wrong?

Actually, this is not a math problem at all, but a mathematical example of the classical magical principle of misdirection. The problem glibly states "which is a total of $27.00; plus the $2.00 the Bell Boy pocketed", which is a meaningless calculation, but serves to distract you from the correct calculation. The correct calculation would be "which is a total of $27.00; including the $2.00 the Bell Boy pocketed". This way you can see that the difference is $25, the price of the room. The misdirection works because the answer it provides is close to something you expect (the original $30).

Another Way of Looking At It...

I could ask why you are adding amounts spent and amounts received. The men spent $27, the bellboy received $2. Let's look at the total amounts spent and received. The men initially spent $10 each, for a total of $30. The hotel received the $30, but only kept $25, returning the other $5, so the hotel received $25. The bellboy received $2 (apparently a carrying charge!), and the men each received a $1 refund (for a total of $3). Seen this way, the men spent $30, and the hotel received $25, the bellboy received $2, and the men received $3, for a total received of $30.

If we wanted to get more complicated, the men spent $30 on the room, which the hotel received. The hotel then spent $5 on the refund, of which the bellboy received $2 and the men received $3. Then the total spent is $35 ($30 by the men and $5 by the hotel), and the total received is $35 ($30 by the hotel, $2 by the bellboy, and $3 by the men as a refund).

Or We Could Look at an Extreme Case...

What happens if we assume the bellboy was completely dishonest, and decided to keep the $5? Then the question would have been:
"But when he does that and gives each man nothing, that means they still paid $10.00 each for the room, which is a total of $30.00; plus the $5.00 the Bell Boy pocketed is a total of $35.00. But they only gave the desk clerk $30.00! Where did the extra $5.00 come from??"
This is either a mathematical proof that crime really does pay, or, more likely, a very suspect statement. That's why the numbers in the original were chosen to make it look like it almost worked. That way you go along with the misdirection, get misled by the wrong calculation, and are confused by the result.

Of course, what actually happened (in our modified case) was that each man paid $10 for the room, for a total of $30.00, of which the hotel got $25 and the bellboy got $5.

In Summary...

This problem is an example of using quick-talking to convince people to accept what you're saying, rather than checking the calculations themselves. If you want another example, take a look on my number tricks page. The trick with cancelling digits uses the same type of misdirection. How well does it work? Over the years, I've tried this trick out on several people, from casual friends to university professors, and have managed to get most of them to at least scratch their heads. Because of the misdirection, they wonder why it works, not whether it works.
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