## Archive for May, 2014

### 2 Truths and a Lie (Mathematician Version)

Here’s the question that started all of the nonsense that follows:

You come to a fork in the road. One fork leads to the village, the other leads to almost certain death. There are three guards stationed at the fork: two always tell the truth, and one always lies. What one question can you ask to one of the guards to find out which fork leads to the village?

There is a truly logical answer to this question, but my favorite answer is:

Did you know they’re giving away free beer in the village?

and then follow all three of them as they sprint toward the village.

A similar question that got me thinking:

A kind but eccentric king has three beautiful daughters. The eldest daughter always tells the truth, the middle daughter always lies, and the youngest daughter will answer any question randomly, either yes or no. To be sure, you would like to marry the one who always tells the truth; but, you are willing to settle for the one who always lies, because at least you’ll always know where you stand. Under no circumstances would you like to marry the crazy one.

The king offers you the hand of one of his daughters in marriage. He allows you to ask one yes/no question of one of the daughters. What question should you ask to ensure that you don’t marry the crazy one?

Variations of this question have been discussed on Straight Dope and xkcd.

And those two questions got me thinking about the icebreaker game *Two Truths and a Lie*, wherein each person at a social gathering tells two truths and one lie about themselves, and the others have to discern fact from fiction.

So I imagined…

What would happen if the most famous mathematicians in history played

Two Truths and A Liewith one another?

The following is what I suspect some of them might say. (The answers follow below.)

**Isaac Newton**

- Newton’s Cannonball is named after one of my thought experiments.
- The city of Newton, MA, is not named after me, but Newton Township, OH, is.
- The Fig Newton, manufactured by Nabisco, is named after me.

**Rene DesCartes**

- I did not get out of bed most days until 11 o’clock in the morning.
- I posited that boiled water freezes more quickly than other water.
- I started college at the age of 10.

**Abraham de Moivre**

- I noted that I was sleeping 15 minutes longer each day, and using that arithmetic progression, I predicted that the day I would sleep for 24 hours would be the exact day of my death — and I was correct.
- I was unable to garner a university post in England, but I was appointed to a Commission of the Royal Society to determine if Newton or Leibniz was the first to discover the calculus.
- I gained great and immediate notoriety for discovering the normal (bell) curve.

**Euclid**

- My name is a Greek word that means “good glory.”
- Abraham Lincoln would often quote me in his speeches.
- I proved the infinitude of primes using a proof by contradiction.

**Gottfried Wilhelm Leibniz**

- I discovered the calculus.
- I invented the first four-function calculator.
- My vast estate was left to my son after my death.

**Leonardo Pisano (Fibonacci)**

- I love rabbits!
- I sometimes used the name Bigollo to refer to myself, which means “good-for-nothing traveler.”
- The 20th century pianist Liberace created his stage name from a contraction of my book title,
*Liber Abaci*.

**Grace Murray Hopper**

- In 1973, I was the first American and the first woman to be elected a Distinguished Fellow of the British Computer Society.
- I invented the computer language COBOL.
- I received 36 honorary degrees.

**Leonhard Euler**

- All of my work, now collected in
*Opera Omnia*, contains over 70 volumes. - In 1735, Guillaume De L’Isle and I prepared a map of the Russian Empire.
- I was the first to use the notation
*f*(*x*) for a function,*e*for the base of natural logs,*i*for the square root of –1, and π for the ratio of circumference to diameter of a circle.

**Diophantus of Alexandria**

- It is believed that I lived to 84 years of age, based entirely on a problem that appeared in a Greek anthology compiled by Metrodorus.
- I was a potato farmer.
- I proved that 24
*n*+ 7 cannot be expressed as the sum of three squares, for integer values of*n*.

**Evariste Galois**

- I was home-schooled until age 12.
- I was killed in a duel, but history is unsure of the other duelist or the reason for the duel.
- I transcribed most of my ideas for what is now called
*Galois theory*the night before the duel.

The third statement from each mathematician was their lie. Below is explication.

Isaac Newton: The Fig Newton is named after the town of Newton, MA, where it was first manufactured.

Rene DesCartes: Actually, he started college at the age of 8.

Abraham de Moivre: In fact, de Moivre’s discovery of the normal curve went almost unnoticed.

Euclid: While many claim that Euclid’s proof of the infinitude of primes uses a proof by contradiction, Michael Hardy and Catherine Woodgold debunk this belief in *Mathematical Intelligencer*, Vol. 31, No. 4, pp. 44–52. Hardy claims that the proof written by Euclid is simpler and more elegant than the proof often attributed to him.

Gottfried Wilhelm Leibniz: He had neither a vast estate nor a son. He was never married, and he died nearly destitute.

Leonardo Pisano (Fibonacci): Though it would be a great piece of trivia, Liberace’s name had nothing to do with Fibonacci. Liberace was a family name; he was born Władziu Valentino Liberace, but he used only his last name on stage.

Grace Murray Hopper: She received at least 37 honorary degrees, perhaps more.

Leonhard Euler: Euler deserves credit for a lot of things, but he does not deserve credit as the first to use π. That distinction belongs to William Jones who used the symbol in 1706.

Diophantus of Alexandria: Though he claimed that 24*n* + 7 cannot be expressed as the sum of three squares, he had no proof of it.

Evariste Galois: The myth that he basically transcribed Galois theory the night before his death is greatly exaggerated. He wrote a lot that evening, but he published three papers in the year before his death, which collectively contained most of his work.

### Peanut Distribution

When we recently bought honey roasted peanuts at the grocery store, Eli speculated that there were 215 peanuts in the jar.

“I think there are less,” Alex said. “My guess is 214.”

“Okay, so now we have to count them,” Eli said.

“No,” I said, explaining that I didn’t want them touching food that others would be eating. I then showed them the back of the jar, which said that one serving contained **about 39 pieces** and the whole jar contained **about 16 servings**. They knew that 39 × 16 would approximate the number of pieces, and they estimated that the jar contained 40 × 15 = 600 pieces.

But then they wanted the actual value, and I wondered how we could use the estimate to find the exact product. More importantly, I wondered if it was possible to find an algorithm that would allow an easily calculated estimate to be converted to the exact value with some minor corrections.

My sons’ estimate used one more than the larger factor and one less than the smaller factor; that is, they found (*m* + 1) × (*n* – 1) to estimate the value of *mn*. A little algebra should help to help to provide some insight.

The product had a value of 600, so further refinement led to:

This led to an algorithm:

- Find an estimate with nice numbers.
- Add 1.
- Add the larger factor.
- Subtract the smaller factor.

This gives 600 + 1 + 39 – 16 = 624. And sure enough, 39 × 16 = 624.

This method works any time you want to find the exact value of a product when the larger factor is one more than a nice number and the smaller factor is one less than a nice number. Just estimate with the nice numbers, then follow the steps. The method can be modified if the larger factor is one less than a nice number and the smaller factor is one more than a nice number:

- Find the estimate.
- Add 1.
- Subtract the larger factor.
- Add the smaller factor.

So if you want to find the product 41 × 14, then the larger factor is one more than 40 and the smaller factor is one less than 15. The estimate is again 40 × 15 = 600.

Then 600 + 1 – 41 + 14 = 574. And sure enough, 41 × 14 = 574.

The same idea can be extended to numbers that aren’t the same distance from nice numbers. But that’s not the point. The intent was not to find general methods for every combination; instead, the hope was to use an easily calculated estimate as the basis for an exact calculation. I’m not sure this method completely succeeds, but it was fun for an afternoon of mental gymnastics.

### Cows and Probability

Bert Tolkamp et al. were awarded an Ig Nobel Prize for answering a question that has long been on the minds of readers of this blog, and likely on the minds of the populace at large:

Are cows more likely to lie down the longer they stand?

I mean, seriously, how many nights have you lain awake pondering that question?

Their research revealed two startling facts:

- The longer a cow has been lying down, the more likely that the cow will soon stand up.
- Once a cow stands up, it’s impossible to predict how long until that cow lies down again.

Fascinating.

I, for one, will rest easier knowing that these questions have finally been answered.

If you suffer from insomnia, the full article may be more valuable than Unisom, chamomile tea, or counting sheep.

You have to wonder if the researchers used a cow-culator to calculate the probabilities. Or perhaps that had to rely on techniques from advanced cow-culus.

Why do milking stools only have three legs?

Because the cow has the udder!

What do Greek cows say?

*μ*.

What do you call a male cow that swallows a hand grenade?

Abominable.What do you call the same cow 5 seconds later?

Noble.

Here are some other mathy cow jokes I’ve posted in the past.