Mathematicians

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:

On a nice day in the 1940’s, three girls go into a hotel and ask for a triple. The manager says sorry, no triples are available, so he puts them in three singles, at $10 each. The girls go up to their rooms.

A few minutes later, a triple frees up, which costs just $25. So the manager, to be a nice guy, decides to move the girls into the triple and refund the $5 difference. He sends the bellboy up to tell the girls of their good fortune and move them into the triple.

While riding up in the elevator, the bellboy thinks to himself, “How can the girls split the $5? $5 doesn’t divide by 3 evenly. I’ll make it easier for them: I’ll give them just $3 — and keep $2 for myself.” So he gave the girls $3 and moved them into the triple.

Everybody was happy. The girls were happy to get refunds. The manager was happy to be a nice guy. And the bellboy was happy to keep $2.

Now here’s the problem: each girl spent $10 and got $1 back, so each girl spent $9. So altogether, the girls spent $9+$9+$9, which is $27, and the bellboy got $2. That makes $29. But we started with $30. What happened to the extra dollar?

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

Here’s the correct answer:

At the end of the story, who has the $30?

The manager has $25, the bellboy has $2, and the girls have $3.

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:

The girls spent a net of $9+$9+$9, which is $27.

$25 of that went to the manager, and $2 went to the bellboy.

Arrangement

Here’s a puzzle:

Arrange 10 coins so they form 5 rows, each containing 4 coins.

“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:

Draw a 5-pointed star. Put the coins at the 10 corners.

Minor formula

Here’s the most famous minor formula:

__ru__

b4i4q qtπ

18

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!

One statistician will claim that the “typical” salary is $1, because it’s the most popular salary: more people received $1 than any other amount. Another statistician will claim that the “typical” salary is $10, because it’s the middle salary: as many people were paid more than $10 as were paid less. The third statistician will claim that the “typical” salary is $222.40, because it’s the average: it’s the sum of all the salaries divided by the number of people.

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.

If the topic is a wage dispute between labor and management, a statistician paid by the laborers will claim that the typical salary is low (just $1); a statistician paid by the management will claim that the typical salary is high ($222.40); and a statistician paid by the arbitrator will claim that the typical salary is reasonable ($10).

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:

There are 3 kinds of lies:

lies, damned lies, and statistics.

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:

Have you stopped beating your wife?

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:

Suppose some numbers are *not*
interesting. For example, suppose 17 is the first number that’s *not*
interesting. Then people would say, “Hey, that’s interesting! 17 has the very
interesting property of being the first boring number!” But then 17 has become
interesting! So you can’t have a first “boring” number, and all numbers are
interesting!

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

I’ll give a surprise test sometime during the semester.

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:

But once you cross “the semester’s last day” off the list of possibilities, you realize the surprise test can’t be “the day before the semester’s last day” either, because the test would be expected then (since the test hadn’t happened already and couldn’t happen on the semester’s last day). Since the test would be expected then, it wouldn’t be a surprise. So cross “the day before last” off the list of possibilities.

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:

But I assure you, there *will*
be a test, and it *will* be a surprise when it comes.

Think about it.

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:

6 is composite because you can get it from multiplying 2 by 3.

9 is composite because you can get it from multiplying 3 by 3.

15 is composite because you can get it from multiplying 3 by 5.

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:

All odd numbers are prime.

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:

1 is prime. 3 prime. 5 is prime. 7 is prime.

9? — no.

11 is prime. 13 is prime.

9 must be just experimental error, so we can ignore it. All odd numbers are prime.

The chemist would rush for results and say just this:

1 is prime, 3 is prime. 5 is prime. 7 is prime.

That’s enough evidence. All odd numbers are prime.

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):

Sure, 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, 13 is prime, 15 is prime, 17 is prime, 19 is prime, all odd numbers are prime!