## Posts tagged ‘proof’

### Guest Blog @ Proofs from the Book

I discovered Guillermo Bautista’s blog *Proofs from the Book* about a month ago, and I wrote about it last week. But just yesterday, Guillermo published a guest post showing the connection between proofs and jokes written by little ol’ me. His blog is worth checking out whether you want to read my post or not.

Paul Erdös once said, “You don’t have to believe in God, but you should believe in *The Book*.” He was referring to a mystical book in which the most elegant proofs of all theorems appear. *The Book* is the inspiration for the name of Guillermo’s new blog. Unfortunately, Martin Aigner and Günter M. Ziegler had the same inspiration when they published *Proofs from THE BOOK* in 1998. Some may think that this duplication is unfortunate. But I say that sometimes lightning strikes twice. After all, if Newton and Liebniz can both be credited for discovering the calculus, then Guillermo deserves as much credit as others for coming up with this awesome name.

The following are some methods of proof that are covered neither in the book *Proofs from THE BOOK* or at the blog *Proofs from the Book*.

Proof by Obviousness: “The proof is so clear that it need not be mentioned.”

Proof by Lack of Sufficient Time: “Because of the time constraint, I’ll leave the proof to you.”

Proof by Lack of Sufficient Space: “I have discovered a truly marvelous proof of this [theorem], which this margin is too narrow to contain.”

Proof by General Agreement: “All in favor?”

Proof by Imagination: “Wel, let’s pretend it’s true.”

Proof by Necessity: “It had better be true, or the entire structure of mathematics would crumble to the ground.”

Proof by Plausibility: “It sounds good, so we’ll assume it’s true.”

Proof by Intimidation: “Don’t be stupid; of course, it’s true.”

Proof by Accident: “Hey, what have we here?”

### The Math of Maker’s Mark

Last week, Maker’s Mark announced that they would change their recipe. According to COO Bob Samuels, the company was planning to reduce the alcohol content of its bourbon from 45 percent to 42 percent by replacing the removed alcohol with water. But outcry from thousands of bourbon drinkers convinced them to abandon their new 84-proof recipe and continue stocking shelves with 90-proof spirits.

This morning, sports reporters Tony Bruno and Harry Mayes suggested that bourbon drinkers were opposed to the change because they want to get drunk faster.

This got me to thinking about math.

As you know, mathematicians know a thing or two about alcohol.

Where there are four mathematicians, you’ll likely find a fifth.

And so do mathematical objects.

A definite integral walks into a bar. “Ten shots of whiskey, please.”

The bartender asks, “You sure you can handle that?”

“Don’t worry,” says the integral. “I know my limits.”

In the U.S., a “standard” drink is one that contains 0.6 fluid ounces of alcohol, but a standard drink does not necessarily correspond to a typical serving size. In practice, a typical drink of bourbon is a 1.5-ounce pour.

So what does this mean for Maker’s Mark? Sticking with 90-proof bourbon means that a 1.5-ounce drink will contain 0.675 fluid ounces of alcohol, whereas the revised 84-proof bourbon would have contained 0.63 fluid ounces of alcohol. Does that extra 0.045 fluid ounces really make a difference?

A little bit, but not much.

As shown in the table below, a 200-pound man would need to consume 4.76 drinks of 84-proof spirits to reach the legal blood alcohol content limit of 0.08, yet he would only need to down 4.44 drinks of 90-proof spirits. The difference is small. Another third of a drink isn’t much when you’ve already downed 4½.

Number of Drinks to Become Legally Drunk (0.08 BAC) |
|||

Weight (lbs) |
Typical Spirits(80 Proof) |
Proposed Maker’sMark (84 Proof) |
Original Maker’sMark (90 Proof) |

100 | 2.50 | 2.38 | 2.22 |

150 | 3.75 | 3.57 | 3.33 |

200 | 5.00 | 4.76 | 4.44 |

250 | 6.25 | 5.95 | 5.56 |

Note that this chart is for men; women of the same weight would require fewer drinks to reach the same level of intoxication. In addition, time is not reflected in this chart. Because of normal body processes, a person’s BAC is reduced by 0.01% every 40 minutes.

Still, the data seems clear. The revised Maker’s Mark recipe would cause intoxication almost as quickly as the original recipe, so if bourbon fans reacted simply because they want to get drunk faster, well, that seems misguided.

Then again, how many sh*tfaced bourbon drinkers have done this kind of analysis? Probably very few. But that does remind me of a joke.

How many bourbon drinkers does it take to change a light bulb?

Just one. Have him drink an entire bottle, then hold the bulb as the room spins.

Or this one that’s a little more mathy.

How many math department chairs does it take to change a light bulb?

Just one. He holds the bulb, and the world revolves around him.

### Blog: Proofs from the Book

I told a friend that Guillermo Bautista had started an interesting new blog called *Proofs from the Book*. “Why would he do that?” my friend asked. “Proofs are so boring!”

I replied in the only way I knew how. “Well, that’s a given.”

Of course, I was making a joke. But lots of people are like my friend and think that proofs are boring. They don’t see the beauty in proofs, probably because they’ve never been exposed to the beauty. That’s why I’m so excited that Guillermo started this blog. His proofs allows students to glimpse the beauty and elegance of mathematical theorems discussed in school mathematics, whether it’s proving that the square root of 3 is irrational, providing multiple proofs for the sum of the first *n* positive integers, or having a little fun and “proving” that 2 = 1.

Two of my favorite proofs follow.

Theorem.Every positive integer is interesting.

Proof.Assume that there is an uninteresting positive integer. Then there must be a smallest uninteresting positive integer. But being the smallest uninteresting positive integer is interesting by itself. Contradiction!

Theorem.A cat has nine tails.

Proof.No cat has eight tails. Since one cat has one more tail than no cat, it must have nine tails.

An excellent proof relies on mathematical insight. The most exciting moment of my mathematical life occurred while I was walking my dog. Though I had found an answer to the Three Points on a Square problem, I had no proof that it was correct, other than thousands of examples generated by Excel. With no pencil, no paper, and no agenda — just some time to think — an elegant proof came to me as I was picking up feces. (I have no idea what that says about me.)

This is my favorite part of mathematics. I can literally spend hours reworking equations, drawing figures, and thinking about a problem, and I’ll make no progress. Then later, when I least expect it, when I’m freed from the confines of pencil and paper, the solution gently alights in my mind like a butterfly coming to rest on a marigold.

My hope is that everyone has a chance to see as much beauty in mathematics as I have seen, and *Proofs from the Book* is a place where you can take a peek.

The following jokes, taken from *Math Jokes 4 Mathy Folks*, provide proof that math can be funny. Sort of.

Did you hear about the one-line proof of Fermat’s Last Theorem?

It’s the same as Andrew Wiles’ proof, but it’s written on a really long strip of paper.

At a conference, a mathematician proves a theorem.

Someone in the audience interrupts him. “But, sir, that proof must be wrong. I’ve found a counterexample.”

The speaker replies, “I don’t care — I have another proof for it.”

What’s the difference between an argument and a proof?

An argument will convince a reasonable man, but a proof is needed to convince an unreasonable one.

A meek man appeared in a court room, and the judge was incredulous when he read the charges against the man. “Sir,” said the judge, “you’re a well educated man. How did you end up here?”

“I’m a mathematical logician, dealing in the nature of proof.”

“Yes, go on,” said the judge.

“Well, I was at the library, and I found the books I wanted and went to take them out. The librarian told me I had to fill out a form to get a library card, so I filled out the forms and got back in line.”

“And?” said the judge.

“And the librarian asked, ‘Can you prove you’re from New York City?’ So I stabbed her.”

### Lead by Counterexample

A wonderful, handwritten sign hangs on the door to my friend’s office:

There is no way of falsifying, “Unicorns exist.”

This is a good example of the nature of mathematics. Although providing just one counterexample is proof that something is not true, the inability to provide a counterexample is not proof that something is true. Understanding this concept is fundamental to understanding mathematics. Perhaps this guy could use a refresher…

At a conference, a mathematician proves a theorem. A young woman in the audience interrupts him. “Excuse me, sir, but I believe that your proof must be wrong. I have found a counterexample to your theorem.”

“Not a problem,” the speaker replies. “I have another proof.”

### Fair and Square Solution

A few days ago, I posted the following problem:

Three points are randomly chosen along the perimeter of a square. What is the probability that the center of the square will be contained within the triangle formed by these three points?

In that post, I mentioned that I had an elegant solution. If you want to think about the problem a little before seeing my solution, check out the Three Points page at the MJ4MF website. I created a really cool Excel file that allows you to explore 100 cases at a time. I think it’s worth a look.

But if you’re not interested in any of that, here’s my solution…

Without loss of generality, choose any two points along the perimeter. Call them A and B. Then, construct segments AD and BC that pass through point P, the center of the square.

This gives four cases:

- If A and B are the two chosen points, then the third vertex must be chosen along segment CD to form a triangle that contains point P.
- If B and D are the two chosen points, then the third vertex must be chosen along segment CA.
- If D and C are the two chosen points, then the third vertex must be chosen along segment AB.
- If C and A are the two chosen points, then the third vertex must be chosen along segment BD.

What we’ve now done is selected two sets of points in **four** different ways, and the portions along the segments where the third point could be chosen collectively comprise the **entire perimeter**. Consequently, the probability that a point along the perimeter is 1, but this is divided among 4 cases, so the probability that a randomly selected triangle will contain the center is **1/4**.

As a footnote, points A and B were chosen arbitrarily, so there is no loss of generality.