## Archive for February, 2013

### Math Dot-to-Dot

Connect-the-dots puzzles usually aren’t very interesting. The purpose of these puzzles is to teach kids the counting numbers, or the alphabet, or something else that occurs in a particular order. Consequently, a dot-to-dot puzzle often contains an image that can be identified *before* the dots are connected, and the image then serves as a scaffold to help students learn the items to be ordered. For adults who know how to count and can identify the image immediately, what’s the point?

For instance, can you identify the shape that will be formed by this connect-the-dots puzzle?

If you had trouble with that, you may want to stop here.

Connect-the-dots puzzles without numbers, however, can yield interesting results. For instance, if the numbers and segment are removed from the puzzle above, the dots can be connected to form more whimsical shapes. With a little creativity, it can result in a fun picture.

So, here’s the challenge I now pose to you:

Using either set of dots below, connect them in any way you like. Allow lines to cross one another, use curves, use only some of the dots, whatever. Be creative.

Then upload your image(s) to Math Dots on Flickr, or post them in the comments below.

**Option 1:**

**Option 2:**

### 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?”

### Math Limerick Problems

Albert Einstein said that “pure mathematics is, in its way, the poetry of logical ideas.” That may or may not be true, but all I know is that math poems are pretty awesome.

There are lots of math limericks on the web. One of my favorites:

A topologist’s child was quite hyper,

Till it wore a Möbius diaper.

The mess on the inside

Was thus on the outside,

And it was easy for someone to wipe her.

Fred Tofts, who claims to not be a mathematician but loves mathematics, recently shared a different kind of math limerick with MJ4MF. His five-line creation was not meant to deliver a punch line; rather, it presented a problem. As a comment to my blog interview with Colin Adams, he wrote, “I have not written any math jokes but have written many math limericks,” and then shared the following:

A dog’s at one end of a log;

At the opposite end is a frog.

Six feet from the frog

And eight feet from the dog

Is a right angle. How long’s the log?

I do hope that the good Mr. Tofts will share a few more of his creations with us!

The following is more of a truth than a puzzle, but fun nonetheless.

Pick a number 1 to 9, I plea,

Then multiply by 15,873.

And again times seven,

The product to leaven;

Your number will repeat six times — you’ll see.

**Do you have any math limerick problems worth sharing?**

### Why Did Vi Hart Go to Khan Academy?

I love Vi Hart. And with over 300,000 subscribers and 25 million views on her YouTube channel, I’m clearly not alone.

But perhaps you don’t know who she is. Maybe you’ve been living under a rock. Maybe you’re still using dial-up. Or maybe you’ve just been posing as a mathy folk, only visiting this blog because you think the author is hot. (Of course, you’d be correct in your assessment, but you shouldn’t let hot authors guide your tour through the blogosphere.)

If you don’t know who Vi Hart is, you can check out her Binary Trees video below (from her now famous Doodling in Math Class series).

Pretty awesome, huh?

In the video, she makes the following statement:

…if the [math] curriculum wasn’t so appalling and the teaching methods weren’t so atrocious, you wouldn’t have to entertain yourself with these stories and games.

She also implies that many math classes are

…fuzzy, unfocused, and altogether not very good.

Some educators don’t like these videos. Some don’t like that a brash, young woman is criticizing what they do and how they do it. Some find her statements offensive.

Not me.

I think she’s spot on.

Too many math classrooms still look like the math classrooms of yesteryear, devoid of excitement and technology and filled with endless hours of meaningless practice.

But here’s where I have trouble. On the About Vi page of her site, she says:

I am now a full-time mathemusician at Khan Academy! It’s pretty exciting.

If she is truly opposed to appalling curriculum, why would she work for a company that creates the video version of a 1950’s textbook?

Maybe I’m being too harsh. But I don’t think so. Though she now creates recreational math videos for Khan Academy that are awesome, the vast majority of videos on the site are nothing more than math lectures of topics that probably should have been removed from the curriculum years ago. When I asked a colleague his thoughts, he had this to say:

Vi’s videos show such polish and cleverness, while Khan’s were so obviously made by someone who just took an exercise from a textbook and sat down at a computer and improvised. About the only thing [Khan Academy] has going for it is that it’s free. I suppose it can have some good use in getting kids an opportunity to learn and practice skills they need, but having them practice skills for no particularly good reason… it’s just reinforcing everything that’s wrong with math education.

In her Binary Trees video, Vi Hart makes fun of the boring presentation of exponential functions that typically occurs in math classes. Yet the Khan Academy video Exponential Growth Functions uses the same examples and “atrocious teaching methods” that would be found in many of the math classes that are “fuzzy, unfocused, and altogether not very good.”

So, what’s up, Vi? How can you rail against bad teaching but then go to work for a place that delivers bad teaching in spades? Your work is amazing, and you had such an opportunity. I hope your intent is to make change from within rather than assimilate.

**How do you feel about Vi Hart’s move to Khan Academy? And what do you think about Khan Academy in general?**

### 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.

### Interesting Exponent Problem

In the expression 3*x*^{2}, the 2 is called an *exponent*. And in the expression 4*y*^{3}, the 3 is called the *y*‑ponent.

Speaking of exponents, here’s a problem you’ll have no trouble solving.

If 2

^{m}= 8 and 3^{n}= 9, what is the value of m · n?

This next one looks similar, but it’s a bit more difficult.

If 2

^{m}= 9 and 3^{n}= 8, what is the value of m · n?

One way to solve this problem is to compute the values of *m* and *n* using a logarithm calculator, though there’s an algebraic approach that’s far more elegant. (I’ll provide the algebraic solution at the end of this post, so as not to spoil your fun.)

Take a few minutes to solve that problem. Go ahead, give it a try. I’ll wait for you.

[…]

Ah, good. Glad you’re back.

Are you surprised by the result? I sure was.

It made me wonder, “If the product of the two numbers is 72, will it always be the case that *m* · *n* = 6?”

It only took me a minute to realize that the answer is no, which can be seen with numerous counterexamples. For instance, if 2^{m} = 64, then *m* = 6. This would mean that 3^{n} = 9/8, because 64 × 9/8 = 72, but it would also mean that *n* = 1, since *m* · *n* = 6. That is obviously a contradiction, since 3^{1} ≠ 9/8.

If the product of the two numbers is held constant at 72, then the problem looks something like this:

If 2

^{m}= k and 3^{n}= 72/k, what is the value of m · n?

The following graph shows the result for 1 < *k* < 72.

So that leads to a couple of follow-up questions.

For what integer values of p, q will p · q = 72 and m · n = 6, if 2

^{m}= p and 3^{n}= q?

If 2

^{m}= p and 3^{n}= q, for what integer values of p, q will m · n = 6 when m, n are not integers?

The answers, of course, are left as an exercise for the reader.

Here’s the algebraic solution to the problem above.

Since 2^{m} = 9, then 2^{mn} = 9^{n} = 3^{2n} = 8^{2} = 2^{6}, so *mn* = 6.

My colleague Al Goetz solved it as follows.

Since

Then

### 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.”

### Math Jokes for Pitchers and Catchers

Pitchers and catchers in Major League Baseball report to spring training this week. Here’s a math joke for baseball players. (And while I’m not certain who said this originally, it’s a solid bet that this quote comes from Dave Barry, though I haven’t been able to confirm that.)

Everyone knows that if a mathematician had to choose between solving a difficult story problem and catching a fly ball, he would surely solve the problem without thinking twice about whether the Infield Fly Rule was in effect.

And since we’re talking baseball…

What do you do with an elephant with three balls?

Walk him, and pitch to the rhino.

Here are some baseball and math trivia questions for you **[updated 2/10/2013]**, some based on problems from Erich Friedman’s Baseball Puzzles page:

- In the bottom of the ninth inning in a game with no score, the bases are loaded. The batter hits a ball that rolls into the right field corner. Before the ball is thrown home, all four base runners cross the plate. What is the final score of the game?
- If the visiting team scores 2 runs per inning, and the home team scores 3 runs per inning, what is the final score of a nine-inning game?
- A player has four at-bats in a game, and he got a hit in his last of these at-bats. His batting averages for the season (rounded to three decimal places, as usual) at three different times during the game do not have any digits in common. What was his batting average at the end of the game?
- What is the minimum number of games a Major League Baseball team must win to make the playoffs?

Answers

- The final score is
**1-0**. Once the first player touches the plate, the home team wins, and the game is officially over. - The final score is
**24-18**. With a lead in the middle of the ninth inning, the home team does not bat in the bottom of the ninth. - At the end of the game, his average was
**.409**. He was 5 for 18 (.278) at the beginning of the game, 6 for 19 (.316) after one at-bat, and 9 for 22 (.409) after three more at-bats. - A team only has to win
**28 games**. There are four teams in the American League West division, and the requirement in Major League Baseball is that every team must play either 18 or 19 games against the other teams in their division. That means that each of these four teams could play as few as 54 divisional games. If each of them loses the 108 non-divisional games that they play, then the four teams could conceivably finish with 28, 27, 27, and 26 wins, respectively. Under this scenario, the division winner is the team with just 28 victories.

There are many who complain that, because of the Electoral College, it’s possible for a candidate to be elected President with less than 30% of the popular vote. Yet this situation is worse — a team could make the playoffs by winning just 17% of its games. (Of course, the Presidential election has slightly more consequence. Maybe.)

### One Joke Per Cent

So, I understand what Baltimore Ravens coach John Harbaugh meant when he said:

I never thought you could feel 100% elation and 100% devastation at the same time. But I learned tonight you can.

But it sure sounds to me like he has twice as much capacity for emotion as the rest of us. It reminds me of Yogi Berra’s famous quote:

Baseball is 90% mental, and the other half is physical.

Or the anonymous quote about our favorite subject:

Mathematics is 50% formulas, 50% proofs, and 50% imagination.

Dare to guess what percent of Americans wouldn’t be able to identify the math errors in those statements?

Here’s a good old-fashioned math joke involving percents:

What’s a proof?

One-half percent of alcohol.

And a slightly longer one:

“Statistics is wonderful!” said a statistician.

“How so?” asked his friend.

“Well, according to statistics, there are 42 million alligator eggs laid every year. Of those, only about 50% hatch. Of those that hatch, 75% are eaten by predators in the first 36 days. And of the rest, only 5% get to be one year old for one reason or another.”

“What’s so wonderful about that?”

“If it weren’t for statistics, we’d be up to our asses in alligators!”