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:

Wolfram Computation Times

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 = abx + c using the world’s best graphing calculator from Desmos, I found a model that may approximate the data a little better:

y = 57 · 0.65x + 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 ex,” says the exponential confidently.

“That’s all well and good,” replies the derivation, “but who says I differentiate with respect to x?”

September 11, 2014 at 9:11 am 2 comments

Math Jokes from Reader’s Digest

All of the following jokes were borrowed from Reader’s Digest, which I’m sure they borrowed from elsewhere.

Did you hear about the mathematician who’s afraid of negative numbers?
He’ll stop at nothing to avoid them.

How easy is it to count in binary?
It’s as easy as 01 10 11.

A Roman walks into the bar, holds up two fingers, and says, “Five beers, please.”

How many bananas can you eat if your stomach is empty?
Just one. Then it’s not empty anymore.

What do you call a number that sleepwalks?
A roamin’ numeral.
(And a nun who sleepwalks?
A roamin’ Catholic.)

Knock! Knock!
Who’s there?
Convex.
Convex who?
Convex go to prison!

September 8, 2014 at 5:45 pm 1 comment

Ring Me Up!

Cash RegisterWhen my college roommate contracted crabs, he went to CVS to buy some lice cream. As you can imagine, he didn’t want to announce to the world what he was buying or why, so he put the box on the counter with a notepad, a bottle of aspirin, a pack of cigarettes, a bag of M&M’s, and a tube of toothpaste — hoping the cream would blend in. The attractive co-ed clerk at the register rang him up without a second look.

As he walked out of the drug store thinking he had gotten away with it, he opened the cigarettes, put one to his lips, and realized he had nothing with which to light it. He returned to the checkout and asked the clerk for a pack of matches.

“Why?” she asked. “If the cream doesn’t work, you gonna burn ‘em off?”

Ouch.

My luck with clerks wasn’t much better. At a grocery store, I placed a bar of soap, a container of milk, two boxes of cereal, and a frozen dinner on the check-out counter. The girl at the cash register asked, “Are you single?”

I looked at my items-to-be-purchased. “Pretty obvious, huh?”

“Sure is,” she replied. “You’re a very unattractive man.”

I did, however, have an exceptional experience at a convenience store. This is what happened.

I walked into a 7-11 and took four items to the cash register. The clerk informed me that the register was broken, but she said she could figure the total using her calculator. The clerk then proceeded to multiply the prices together and declared that the total was $7.11. Although I knew the prices should have been added, not multiplied, I said nothing — as it turns out, the result would have been $7.11 whether the four prices were added or multiplied.

There was no sales tax. What was the cost of each item?

As you might have guessed, that story is completely false. (The one about me being called ‘unattractive’ is a slight exaggeration. The one about my roommate, sadly, is 100% true.) The truth is that I learned this problem from other instructors when teaching at a gifted summer camp.

It may not be true. It is, however, one helluva great problem.

But it has always bothered me that the problem is so difficult. I’ve always wanted a simpler version, so that every student could have an entry point. Today, I spent some time creating a few.

Use the same set-up for each problem below… walk into a store… take some items to check-out counter… multiply instead of add… same total either way. The only difference is the number of items purchased and the total cost.

I’ve tried to rank the problems by level of difficulty. Below, I’ve given some additional explanation — but not the answers… you’ll have to figure them out on your own.

  • (trivial) Two items, $4.00.
  • (easy) Two items, $4.50.
  • (fun) Two items, $102.01.
  • (systematic) Two items, $8.41.
  • (perfect) Three items, $6.00.
  • (tough) Three items, $6.42.
  • (rough) Three items, $5.61.
  • (insane) Four items, $6.44.
  • (the one that started it all) Four items, $7.11.

Editor’s Notes

trivial — C’mon, now… even my seven-year-old sons figured this one out!

easy, fun, systematic — All of these are systems of two equations in two variables. Should be simple enough for anyone who’s studied basic algebra. All others can use guess-and-check.

perfect — Almost as easy as trivial, and the name is a hint.

tough — But not too tough. Finding one of the prices should be fairly easy. Once you have that, what’s left reduces to a system of equations in two variables.

rough — Much tougher than tough. None of the prices are easy to find in this one.

insane — Gridiculously hard, so how ’bout a hint? Okay. Each item has a unique price under $2.00. If you use brute force and try every possibility, that’s only about 1.5 billion combinations. Shouldn’t take too long to get through all of them…

the one that started it all — As tough as insane, and not for the faint of heart. But no hint this time. Good luck!

 

September 5, 2014 at 7:11 am Leave a comment

A Gridiculously Clever Blog Post

Do you know what the following graph represents?

Sine on the Dotted Line

Sine on the dotted line.

If you tell that joke to the right audience, you’ll likely hear a triggle. (If you tell it to the wrong audience, you’ll likely hear the sound of tomatoes whizzing past your head.)

Triggle is a portmanteau, a combination of two or more words and their definitions.

trigonometry + giggle = triggle

In a similar vein, when the expression

13 + 5 · 0 – 4

is simplified to

13 – 4,

you might say that it has suffered from zerosion — the removal of a term because of multiplication by zero.

The following portmanteaux may be useful for your next math discussion.

bi·sect·u·al
adjective
attracted to both halves of an angle

grid·ic·u·lous1
adjective
inviting derision on the coordinate plane

cha·rad·i·us
noun
a segment from the center to the circumference based on false pretenses

bi·zarc
noun
an unusual curve

graph·ish
adjective
diagrammatically disreputable

sub·line
adjective
inspiring awe in only one dimension

trig·a·ma·role
noun
a complicated and annoying trigonometric process, such as verifying that
cot x + tan x = sec x · csc x


1 It came to my attention after the publication of this post that Gridiculous is (a) a trivia game developed for Windows 8 and (b) an HTML5 responsive grid boilerplate (though the link to the site seems not to be working).

September 2, 2014 at 4:07 pm Leave a comment

Easiest KenKen Ever?

SIgmund FreudSaying that I like KenKen® would be like saying that Sigmund Freud liked cocaine. (Too soon?) ‘Twould be more proper to say that I am so thoroughly addicted to the puzzle that the length of my dog’s morning walks aren’t measured in miles or minutes but in number of 6 × 6 puzzles that I complete. (Most mornings, it’s two.) Roberto Clemente correctly predicted that he would die in a plane crash; Abraham de Moivre predicted that he would sleep to death (and the exact date on which it would occur… creepy); and I am absolutely certain that I’ll be hit by oncoming traffic as I step off the curb without looking, my nose pointed at a KenKen app on my phone and wondering, “How many five-element partitions of 13 could fill that 48× cell?”

I am forever indebted to Tetsuya Miyamoto for inventing KenKen, and I am deeply appreciative that Nextoy, LLC, brought KenKen to the United States. How else would I wile away the hours between sunrise and sunset?

I am also extremely grateful that the only thing Nextoy copyrighted was the name KenKen. This allows Tom Snyder to develop themed TomToms, and it allows the PGDevTeam to offer MathDoku Pro, which I believe to be the best Android app for playing KenKen puzzles.

The most recent release of MathDoku has improved numerical input as well as a timer. Consequently, my recent fascination is playing 4 × 4 puzzles to see how long it will take. A typical puzzle will take 20‑30 seconds; occasionally, I’ll complete a puzzle in 18‑19 seconds; and, every once in a while, I’ll hit 17 seconds… but not very often.

Today, however, was a banner day. I was in a good KenKen groove, and I was served one of the easiest 4 × 4 puzzles ever. Here’s the puzzle:

Easy 4x4 KenKen

And here’s the result (spoiler):

Easy KenKen

4 × 4 KenKen solved in 15 seconds

The screenshot shows that I completed the puzzle in just 15 seconds. And it’s not even photoshopped.

This puzzle has several elements that make it easy to solve:

  • The [11+] cell can only be filled with (4, 3, 4).
  • The [4] in the first column dictates the order of the (1, 4) in the [4×] cell.
  • The (1, 4) in the [4×] cell dictates the order of the (1, 2) in the [3+] cell.

After that, the rest of the puzzle falls easily into place, because each digit 1‑4 occurs exactly once in each row and column.

What’s the fastest you’ve ever solved a 4 × 4 KenKen puzzle? Post your time in the comments. Feel free to post your times for other size puzzles, too. (I’m currently working on a 6 × 6 puzzle that’s kicking my ass. Current time is 2:08:54 and counting.)

August 18, 2014 at 8:11 am Leave a comment

Think of a Number

I love to create math games almost as much as I love to play them.

Calculation NationMy favorite professional project was leading the development of Calculation Nation. And my favorite game on the site is neXtu, though other games on the site may promote more sophisticated mathematical thinking.

I have many reasons to love my wife, not least of which is her creation of the game Dollar Nim. While I can’t take credit for the rules, I will take credit for its analysis and its popularization. (What do you call a wife who makes up a game that gets you a publication credit? A keeper!)

DragonboxRecently, I’ve been frustrated by the lack of games for teaching algebra. I’ll give props to the good folks at Dragonbox, which uses a game environment to teach algebra. But I’m not yet convinced that it leads to deep algebraic understanding; even they admit “to transfer to pencil and paper, children must be explained how to rewrite equations line by line.” They also claim that “in-house preliminary tests indicate a very high level of transfer to pencil and paper,” but that’s the fox watching the henhouse.

So I’ve been thinking about games I can play with my sons that will allow them to engage in algebraic thinking. But I don’t want them to know they’re engaging in algebraic thinking. I have two criteria for all math games:

  • The game mechanics depend on mathematics. The math is not tangential to the game; it is the game.
  • Kids don’t realize (or at least they don’t care) that it’s a math game, because it’s fun.

It pains me to write that second criterion, because math is fun. But I know not everyone shares that opinion. So I do my best to disguise any math learning in the game and then, when they least expect it — BOOM! — I drop the bomb and show them what they’ve learned.

So here’s a game I recently devised.

  • Player A chooses a number.
  • Player B chooses two operations for Player A to perform on the number.
  • Player A performs those operations and then tells the result to Player B.
  • Player B then tries to identify Player A’s number.

These rules leave something to be desired, since Player B could simply ask A to “multiply by 1″ and then “add 0,” in which case finding A’s number would involve no work whatsoever. To be a stickler, an additional rule could impose that either addition or subtraction can be used exactly once and that no operation can involve either 0 or 1. In a middle school classroom, I suppose I would state such a rule explicitly; for playing this game with my seven-year-old sons, I opted not to.

We played this game three times on the car ride to school yesterday. One game went like this:

  • I thought of a number (14).
  • Eli asked me to add 3 to my number.
  • Alex asked me to multiply by 3.
  • I told them the result: 51.

Eli then guessed that my number was 16. He had subtracted 3, then divided by 3.

“No!” said Alex. “You added 3 first, so you need to subtract 9.”

“Why 9?” Eli asked. “Daddy only added 3.”

“But he multiplied by 3, so if you subtract first, you have to subtract 3 × 3.”

Eli then realized that my number was 14.

He thought for a second. “Oh,” he said. “I should have divided by 3 first, then subtracted.”

Wow, I thought. This is going even better than I hoped.

Though they didn’t use the proper terminology, the boys had a great discussion about “undoing” operations by performing inverse operations in reverse order. In 10 minutes, they taught themselves how to solve a two-step equation:

3x + 3 = 51

Grace Kelemanik once said that she knew she was being effective when she didn’t have to say a word. She’d watch from the back of the room as students carried the conversation and guided one another to correct mathematical thinking.

I will never claim to be half the educator that Grace Kelemanik is. But yesterday morning, I was pretty darn effective.

I’d love to hear about math games you’ve played with kids, whether you invented them or not.

August 15, 2014 at 11:43 pm 4 comments

Things I Learned on a Boston Duck Tour

Dirty Waters

ConDUCKtor Dirty Waters

Dirty Waters led our Boston Duck Tour yesterday and told us a little about himself:

I’m wicked smaht. In fact, I was valedictorian of my high school. Of course, I was homeschooled… but my mom says it still counts.

Dirty was a veritable fountain of math-related trivia. For instance, he told us that the movie Good Will Hunting, in which Matt Damon roams the halls of Ford Building at MIT solving difficult math problems, wasn’t actually filmed at MIT. Rather,

The hallway scenes were filmed at Beacon Hill Community College… and let’s be honest, anyone can answer the math questions that are asked there.

Incidentally, the math problem that Damon solved involved drawing all the homeomorphically irreducible trees of degree 10. While I don’t know how well the typical BHCC student might react to this problem, I do know that my seven-year-old sons were able to solve it — once I helped them understand what a homeomorphically irreducible tree was.

Irreducible Tree of Degree 10

Irreducible Tree

We also learned the following non-math trivia about Paul Revere:

  • Paul Revere didn’t actually make it to Concord. He was captured by the Redcoats and sang like a songbird — he divulged the entirety of the colonists’ plans.
  • He didn’t yell, “The British are coming! The British are coming!” That would have made no sense. At the time of his midnight ride, all of the colonists considered themselves British. Instead, he probably yelled, “The Regulars are coming!” a term used to describe British soldiers.
  • That’s not Samuel Adams on the front of a Sam Adams bottle. It’s Paul Revere, who was much more handsome than Adams.

This made me realize that a lot of the things we learn(ed) in school are complete bullshit:

  1. Paul Revere informed the folks in Concord that the British were coming. In fact, Samuel Prescott was the only rider to reach Concord. A third rider that night, William Dawes, accompanied Revere and Prescott, but he was thrown from his horse and walked back to Lexington.
  2. Humans have five senses (sight, smell, touch, taste, hearing). Actually, no… most social scientists also include pain, hunger, thirst, pressure, balance, acceleration, and time, among others.
  3. Sentences cannot end with prepositions. Not true, and sometimes you’ll sound like Yoda if you try to do otherwise (e.g., “Rained out was the baseball game”). The classic joke is, “What is a preposition? A preposition is a word one must never end a sentence with.”
  4. Division by zero is impossible. It’s not impossible; it’s just a bad idea. Weird stuff happens when you divide by zero, and it’s easier to avoid it by calling the action “undefined.”
  5. Chameleons change color to blend in. ‘Twould be awesome were it so, but they actually change color to communicate. While you might flip someone the bird to let them know you’re unhappy, a chameleon would just change to a darker color.
  6. Columbus thought the world was flat. No, he didn’t, and neither did most educated people at the time. Columbus’s mistake was actually underestimating the size of the Earth. He was lucky to have found the West Indies, lest he and all of his crew would have died of starvation.

Why do these inaccuracies persist? I suspect most of the errors are legacy content from hundred-year-old curriculum; the alternative is that it’s willful deceit on the part of educators, and that’s hard to swallow.

What other complete bullshit is still perpetuated in American classrooms?

Leave a comment.

August 7, 2014 at 8:07 am 2 comments

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About MJ4MF

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

MJ4MF (offline version)

Math Jokes 4 Mathy Folks is available from Amazon, Borders, Barnes & Noble, NCTM, Robert D. Reed Publishers, and other purveyors of exceptional literature.

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