A Mathematics Problem

Welcome to the first issue of “Intractable Problems,” a column designed to stimulate and challenge your thinking skills. Its purpose is to present examples of interesting and curious problems which we may face every time we enter into a business engagement.

What, you may ask, is an intractable problem? Well, for those of us without a dictionary at hand, an intractable problem is one which is quite difficult to solve, perhaps even impossible. You know, the types of problems that each and every one of us face every day. The problems that we all excel at, because that’s what we do best!

Intractable problems exist everywhere. I intend to focus on problems in language, math, logic, information theory, and computer science. A counterpoint to sales and marketing, for readers of a different ilk.

To get us started, I thought we would begin with a math problem. This problem may be quite useful for those of us who try to make the numbers add up every quarter. [1]

One day, three people came into a hotel to rent a room for the night. The desk clerk charged them $60 for the room. Later, after the people had gone up to their room, the desk clerk realized that he had overcharged them $5 for the room. So, he gave the bellhop the five dollars and asked that he return it to the three guests in their room.

Now, the bellhop was not too bright, and he wondered how he would divide the $5 evenly among the three guests. With a brilliant flash of insight, he decided to give the three guests one dollar each, and to pocket the remaining two dollars himself.

Later that night the bellhop began to think … Each guest had actually paid $19 for the room ($20 less the $1 returned to them). Thus, since the $57 paid plus the $2 in his pocket gives $59, where did the missing dollar go?

To make this problem more interesting, I will consider giving the missing dollar to the third person who sends me a letter with the correct explanation. See, you don’t have to be first to win!

Next issue, I will show you that there are some real problems which just can’t be solved – in any known or conceivable way! (Computers can’t do everything.)


1. Paulos, J. A Mathematician Reads the Newspaper, HarperCollins Publishing, 1995, pp. 86.