## True Inequalities

It’s true that Bertrand Russell once stated he could prove anything, given that 1 + 1 = 1. What’s likely not true is that someone challenged Russell to prove that he was the Pope, and he responded by saying, “I am one. The Pope is one. Therefore, the Pope and I are one.”

Whatever. Even apocryphal, it’s a fun story. Who needs truth, anyway?

Ask the poet (Keats) who said that what the imagination seizes as beauty must be truth.

He might also have said that what the hand seizes as a ball must be truth, but he didn’t, because he was a poet and preferred loafing about under trees with a bottle of laudanum and a notebook to playing cricket, but it would have been equally true.

— Douglas Adams, Dirk Gently’s Holistic Detective Agency

The following inequalities are — under some circumstances — true.

–

**1 + 1 = 1**

See above. (Were you even paying attention?)

–

**1 + 3 = 1**

This inequality comes from an athletic shirt that I own. What happens when one large, hungry fish meets three little fish? One large fish leaves with a full belly. (In case you can’t see it in the picture, there are four sets of small fish bones in the big fish’s belly.)

–

**10 + 10 = 100
**

Binary much?

–

**1/10 = 20%**

Middle school teachers will cringe at seeing this seemingly incorrect fraction-to-percent conversion, but it’s true if you’re looking at a nutrition label. Eat 10g of low-fat Swiss cheese with 1g fat and 9g protein, and 20% of your calories come from fat.

–

**10 + 4 = 2**

On a calculator? No. On a clock? Yes. Move 4 hours past 10 o’clock, and it’s 2 o’clock.

–

**1/2 + 1/3 = 2/5**

More for middle-school teachers to cringe about. But if you play sports and want to compute your shooting, passing, or batting average, this equation is totally legit.

## Infinite Integer Triangles

Here’s an interesting question.

Given the side of a triangle with integer length, what is the set of all points in the plane for which the other two sides will also have integer lengths?

And by *interesting*, I mean that the answer wasn’t immediately obvious to me.

So I drew a segment 5 units long in Geometer’s SketchPad, created a bunch of concentric circles with integer radii and centers at the endpoints of the segment, identified the intersection points of those circles, and finally hid the circles. The result was the following beautiful image:

And by *beautiful*, I mean that the result is, well, beautiful. At least to a math dork. If this had been painted by Van Gogh, it would have been called **Triangle in a Starry Night**. (Okay, maybe not.)

The triangle indicated by the dashed lines is the famous 3-4-5 right triangle. The points in the upper right and upper left corners yield the less well known but similarly intoxicating 5-5-8 triangle. If the limitations of the web allowed this image to extend infinitely in all directions, the result would be infinite beauty. Alas, reality confines us.

I have an infinity of jokes that deal with triangles and circles, but I’ll only share a subset of them here.

What did the triangle say to the circle?

Your life is pointless.Why don’t circles hang out with ellipses?

Too eccentric.What did the hypotenuse say to the other two sides?

Nice legs!Where do circles and ellipses spend their vacations?

Coney Island.What’s a circle?

A round, straight line with a hole in the middle.What did the circle say to the tangent line?

Stop touching me!

## Eat, Sleep, Do Math!

The Golden Rule of Food Shopping:

Never shop for groceries when you’re hungry.

Corollary for Mattress Shopping:

Never shop for a mattress when you’re tired.

When buying a mattress, Consumer Reports recommends that you lie down on “lots of mattresses” in the store and spend at least 15 minutes on each mattress — five minutes lying on each side, and another five minutes on your front or back, depending on your sleeping preference. I’m not certain what number is implied by “lots of mattresses,” and I’ve never been very good at math, but if you try out 6-8 different mattresses for 15 minutes each, plus some chit-chat and the requisite haggling with a salesperson, you’re trip to the mattress store is going to last at least an entire afternoon, maybe more.

They also recommend that you wear loose-fitting clothes, so I donned a smoking robe and slippers. Our family then headed to Sleepy’s.

The first mattress I tested was too firm. It took far less than 15 minutes to eliminate it as a possibility.

The second mattress I tested, however, was damn near perfect. I rested on my left side for five minutes, and it felt very good. I then rolled over to my right side… and I fell asleep. Not sure what to do, my wife did what any dedicated wife would do — she left. She and the kids walked to the grocery store, and when they returned 35 minutes later, I was rousing from my slumber.

“This is the one,” I said.

“Yeah,” she said. “No shit.”

I did not need to test any more. That said, the one I liked was far from the cheapest one on the showroom floor. Consequently, haggling ensued. As I was asking for a 25% discount and the salesperson was countering with, “How ’bout I throw in a free pillow?” my sons were inspecting a poster in the store:

The intent of the poster, of course, is to show that Americans spend 1/3 of their lives in bed. (And, implicitly, to suggest that price should not be a consideration for something you use so often.) But it caused some slight bewilderment for my sons.

Only 21.8 hours are accounted for.

If there had been no category called “Other,” it might not have been so odd. But couldn’t they have included the missing 2.2 hours in “Other”? Unless that time is spent doing something other than “Other,” but I have no idea what that might be.

If this information were represented as a pie chart, it might look like this:

The source of the statistics, according to a footnote on the poster, is the Bureau of Labor Statistics. But that doesn’t seem true. At the top of this data table from BLS, the sum total of all activities is 24.00 hours.

My job is done here. I’m off to enjoy my new mattress. Good night.

## Periodically Crude

Old farts will know the answer to this old trivia question:

What two letters do not appear on a phone?

And if your phone still looked like this…

then it would be a reasonable question.

But phones don’t look like that anymore. They look like this…

in which case, it’s a really dumb question. (The Q is now attached to 7, and Z hangs out with 9.)

On the other hand, the periodic table looks the same today as when Mendeleev published it in 1869, so the following trivia question may be a bit better:

What two letters never appear in a chemical abbreviation on the periodic table? (I mean anywhere, bitches.)

Shouldn’t be that hard, if you’re willing to take the time to look.

Jessica Lee made headlines back in May when she placed the following quote in her yearbook:

Fluorine uranium carbon potassium bismuth technetium helium sulfur germanium thulium oxygen neon yttrium.

Seems innocuous enough, till it’s translated with the periodic table:

(A line from a Notorious B.I.G. song, for the old farts reading this.)

Are you made of nickel, cerium, arsenic and sulfur? Because you have a…

Or maybe you’re made of copper and tellurium? Because you’re…

If you got nothing better to do today, maybe you could take a ride on a ferrous wheel…

## Ahoy, Matey! Math Jokes Ho!

A **ditloid** is a puzzle in which a fact must be discerned from the numbers and abbreviated letters in the clue. For example, **7 D in a W** is a ditloid for “7 Days in a Week,” and **20 V on a D** is a ditloid for “20 Vertices on a Dodecahedron.”

Here’s a ditloid in honor of International Talk Like a Pirate Day:

15 M on a D M C

If you have trouble, here’s a hint.

Also for International Talk Like a Pirate Day, some mathy pirate jokes.

Teacher: What’s the circumference of a circle?

Pirate: 2π Arrr!How much did the pirate pay to have his ears pierced?

A buccaneer!What has eight legs and eight eyes?

Eight pirates!What grade did the pirate get in math class?

High C’s!

When Apple finally enters the pirate arena… the iPatch.

## Book Review: *365 Things To Make You Go Hmmm…*

Before reading *365 Things That Make You Go Hmmm…*, I hadn’t realized that I’d been on Earth for 1.3 billion seconds, and I never thought about what someone would feel like after spending a day in my mind. That’s the beauty of this incredible book — it asks you to think about things that you’ve probably never thought about before. The questions are great for starting classroom discussions, but they also work well for sparking a conversation between a parent and child, or as an icebreaker at your next social event.

The book contains introspective questions (“What makes you irreplaceable?”), but it also contains math and logic puzzles like the following:

Before this piece of paper was folded over once, it was a capital letter. It wasn’t the letter L — that would be too easy. Which letter was it?

I’m also a big fan of puzzle #110, which starts:

An

antigramis word [or phrase] that when you rearrange the letters you can make a new word or phrase that means something very different — in fact, almost the opposite! For example:.earliest→rise late

It then provides a list of antigrams and asks for the opposite word or phrase. One of the antigrams is:

within earshot

Flummoxed, I looked at the answer in the back of the book, which read:

I won’t hear

I realized immediately that something was wrong. The given answer did not contain enough letters. And then I gasped, because I realized which letters had been omitted:

Wow! I emailed Paul Wrangles (the author) immediately and asked if the answer was given as “I won’t hear” so as to avoid writing “I won’t hear shit,” or if this was simply a typo. He assured me that it was only a typo, and the correct answer is supposed to be:

I won’t hear THIS

Whew!

With that mystery solved, I viewed the other 360 things and thoroughly enjoyed them. My sons and I have been working our way through them, though they’re so addictive, we rarely stop at answering just one a day. We’re hoping for a second volume — we need more questions to last an entire year!

*365 Things That Make You Go Hmmm…* is an amazing resource. Chock full of questions from ordinary to extraordinary, it made my head hurt — but in a good way!

I highly recommend this book for any teacher, parent, or curious individual.

## Exponentially Smarter, Literally

To show my sons what Siri can do, I asked her (it?) the following question:

What is 6 + 4?

Siri told me, “The answer is 10.” But she also provided a bunch of other information pulled from Wolfram Alpha, including the following data:

This data appears to be taken from dissertation research by B. A. Fierman which was furthered by psychologist Mark H. Ashcraft. What it shows is that we get exponentially smarter — or at least faster at calculating — as we get older.

According to Excel, this data can be modeled exponentially by *y* = 8.36 · *e*^{–0.129x}, though this model has obvious limitations. For example, it implies that a one-year-old would be able compute this sum in 7.35 seconds, yet I know no one-year-old who understands addition. Further, it claims that it would take me 0.03 seconds to compute the sum, but I would argue first that I don’t compute the sum, I merely recall it; and second, my reaction time when asked for the sum would be greater than 0.03 seconds.

Playing around with the generic function *y* = *ab*^{x} + *c* using the world’s best graphing calculator from Desmos, I found a model that may approximate the data a little better:

*y* = 57 · 0.65^{x} + 0.9

With this model, it would take a one-year-old 37.95 seconds to compute sum. That’s still not reasonable for any one-year-old that I know, but at least the model says it would take me 0.9 seconds to recall the fact, a far more reasonable estimate than the 0.03 seconds given by the Excel model above.

Interestingly, How To Geek claims that Siri uses Wolfram Alpha for 25% of its searches. Yet if you ask Siri, “What is the meaning of life?” it will respond,

I can’t answer that right now, but give me some very long time to write a play in which nothing happens.

or

Try and be nice to people. Avoid eating fat. Read a good book every now and then, get some walking in, and try to live together in peace and harmony with people of all creeds and nations.

On the other hand, if you ask Wolfram Alpha, “What is the meaning of life?” it will respond,

42.

Proper.

All this talk of exponentials reminds me of a joke.

Q: How do you know that your dentist studied algebra?

A: She tells you that candy will lead to exponential decay.

Perhaps the most famous joke about exponentials is not one of which I’m terribly fond. I share it here only to honor my mission of providing math jokes to the world, not because I think any of you will enjoy it.

Several functions are sitting in a bar, bragging about how fast they go to zero at infinity. Suddenly, one hollers, “Look out! Derivation is coming!” All of the functions immediately cower under the table, but the exponential function sits calmly on the chair.

The derivation comes in, sees the exponential function, and says, “Don’t you fear me?”

“No, I’m

e,” says the exponential confidently.^{x}“That’s all well and good,” replies the derivation, “but who says I differentiate with respect to

x?”