In my former life — before I tried to be a writer or a computer guy — I was a mathematician.

The missing dollar

Math teachers love this puzzle:

Ask your friends that question and see how many crazy answers you get!

Here’s the correct answer:

There’s no logical reason why taking the amount that the girls spent and adding it to what the bellboy received should give any meaningful number. But the nonsense total, $29, is close enough to $30 to be intriguing.

Here’s an alternative analysis:

Arrangement

Here’s a puzzle:

“5 rows of 4 coins” would normally require a total of 20 coins, but if you arrange properly you can solve the puzzle. Hint: the rows must be straight but don’t have to be horizontal or vertical. Ask your friends that puzzle to drive them nuts.

Here’s the solution:

Minor formula

Here’s the most famous minor formula:

Try saying it out loud. It’s about dealing with minors.

Statistics

Courses in statistics can be difficult. That’s why they’re called “sadistics.”

Lies Statisticians give misleading answers.

For example, suppose you’ve paid one person a salary of $1000, another person a salary of $100, another person a salary of $10, and two other people a salary of $1 each. What’s the “typical” salary you paid? If you ask that question to three different statisticians, they’ll give you three different answers!

Which statistician is right? According to the Association for Defending Statisticians (started by my friends), the three statisticians are all right! The most common salary ($1) is called the mode; the middle salary ($10) is called the median; the average salary ($222.40) is called the mean.

But which is the “typical” salary, really? Is it the mode ($1), the median ($10), or the mean ($222.40)? That’s up to you!

If you leave the decision up to the statistician, the statistician’s answer will depend on who hired him.

Which statistician is telling the whole truth? None of them!

A century ago, Benjamin Disraeli, England’s prime minister, summarized the whole situation in one sentence. He said:

Logic

A course in “logic” is a blend of math and philosophy. It can be lots of fun — and also help you become a lawyer.

Beating your wife There’s the old logic question about how to answer this question:

Regardless of whether you answer that question by saying “yes” or “no,” you’re implying that you did indeed beat your wife in the past.

Interesting number Some numbers are interesting. For example, some people think 128 is interesting because it’s “2 times 2 times 2 times 2 times 2 times 2 times 2.” Here’s a proof that all numbers are interesting:

Surprise test When I took a logic course at Dartmouth College, the professor began by warning me and my classmates:

Then he told the class to analyze that sentence and try to deduce when the surprise test would be.

He pointed out that the test can’t be on the semester’s last day — because if the test didn’t happen before then, the students would be expecting the test when they walk into class on that last day, and it wouldn’t be a surprise anymore. So cross “the semester’s last day” off the list of possibilities.

Then he continued his argument:

Continuing in that fashion, he said, more and more days would be crossed off, until eventually all days would be crossed off the list of possibilities, meaning there couldn’t be a surprise test.

Then he continued:

Mathematicians versus engineers

The typical mathematician finds abstract concepts beautiful, and doesn’t care whether they have any “practical” applications. The typical engineer is exactly the opposite: the engineer cares just about practical applications.

Engineers complain that mathematicians are ivory-tower daydreamers who are divorced from reality. Mathematicians complain that engineers are too worldly and also too stupid to appreciate the higher beauties of the mathematical arts.

To illustrate those differences, mathematicians tell 3 tales.…

Boil water Suppose you’re in a room that has a sink, stove, table, and chair. A kettle is on the table. Problem: boil some water.

An engineer would carry the kettle from the table to the sink, fill the kettle with water, put the kettle onto the stove, and wait for the water to boil. So would a mathematician.

But suppose you change the problem, so the kettle’s on the chair instead of the table. The engineer would carry the kettle from the chair to the sink, fill the kettle with water, put the kettle onto the stove, and wait for the water to boil. But the mathematician would not! Instead, the mathematician would carry the kettle from the chair to the table, yell “now the problem’s been reduced to the previous problem,” and walk away.

Analyze tennis Suppose 1024 people are in a tennis tournament. The players are paired, to form 512 tennis matches; then the winners of those matches are paired against each other, to form 256 play-off matches; then the winners of the play-off matches are paired against each other, to form 128 further play-off matches; etc.; until finally just 2 players remain — the finalists — who play against each other to determine the 1 person who wins the entire tournament. Problem: compute how many matches are played in the entire tournament.

The layman would add 512+256+128+64+32+16+8+4+2+1, to arrive at the correct answer, 1023.

The engineer, too lazy to add all those numbers, would realize that the numbers 512, 256, etc., form a series whose sum can be obtained by a simple, magic formula! Just take the first number (512), double it, and then subtract 1, giving a final result of 1023!

But the true mathematician spurns the formula and searches instead for the problem’s underlying meaning. Suddenly it dawns on him! Since the problem said there are “1024 people” but just 1 final winner, the number of people who must be eliminated is “1024 minus 1,” which is 1023, and so there must be 1023 matches!

The mathematician’s calculation (1024-1) is faster than the engineer’s. But best of all, the mathematician’s reasoning applies to any tournament, even if the number of players isn’t a magical number such as 1024. No matter how many people play, just subtract 1 to get the number of matches!

Prime numbers Mathematicians are precise, physicists somewhat less so, chemists even less so. Engineers are even less precise and also less intellectual. To illustrate that view, mathematicians tell the tale of prime numbers.

First, let me explain some math jargon. The counting numbers are 1, 2, 3, etc. A counting number is called composite if you can get it from multiplying a pair of other counting numbers. For example:

A counting number that’s not composite is called prime. For example, 7 is prime because you can’t make 7 from multiplying a pair of other counting numbers. Whether 1 is “prime” depends on how you define “prime,” but for the purpose of this discussion let’s consider 1 to be prime.

Here’s how scientists would try to prove this theorem:

Actually, that theorem is false! All odd numbers are not prime! For example, 9 is an odd number that’s not prime. But although 9 isn’t prime, the physicists, chemists, and engineers would still say the theorem is true.

The physicist would say, slowly and carefully:

The chemist would rush for results and say just this:

The engineer would be the crudest and stupidest of them all. He’d say the following as fast as possible (to meet the next deadline for building his rocket, which will accidentally blow up):