## Posts tagged ‘eli’

### Jokes In Order

“Daddy,” said Eli, “there’s a new math joke we need to tell you.”

“Really?” I said. “Where did you hear this joke?”

“I made it up,” Eli said.

“Well, then, let’s hear it!”

What did 0 say to 10?

Nice one!

Okay, so maybe it’s not a knee-slapper… but I still think it’s pretty cool that my 4-year-old son created a math joke.

I especially liked that he made up his joke about 10 on the 10th of September; more specifically, on the sequential date 9/10/11. Here’s a joke about sequences:

Being without you is like being a metric space in which a Cauchy sequence exists but does not converge.

The date 9/10/11 is also interesting in Roman numerals — it is IX/X/XI, which is a palindrome. How many other sets of three consecutive Roman numerals, when taken in order, form a palindrome?

### Out of Sequence

If you came here looking for a good math joke and were thinking, “My God, I sure hope he didn’t post another story about his sons,” well, now would be a good time to close your browser.

Still with me? Good. Because I’ve just got to tell you about two things that have happened recently with Alex and Eli.

As we were walking through the neighborhood today, I was pointing to things that I thought the boys would enjoy. “Look,” I said as I pointed to the sky. Alex looked up. “An airplane!” he shouted.

I pointed to an automobile on the right. “A red car!” Eli exclaimed.

Then I pointed to a house on our left. Alex announced, “A square Fibonacci number!”

I admit, I didn’t see that coming. I had pointed to a house on our street, and there was a cat on the front porch that I wanted the boys to see. Instead, Alex saw the house number — 144 — and identified it as part of two different sequences. (I don’t think he recognized the lovely coincidence that 144 is the 12th term in both sequences, though I can’t be sure.)

My boys’ love of sequences is a result of teeth brushing. (No, really.) To make sure they brush their teeth long enough, I count while they brush — 15 seconds on the left side, 15 seconds on the right side, 15 seconds for the front teeth, and finally 15 seconds for “all around,” during which they’re supposed to brush the inside parts of their teeth. As you might well imagine, though, I was getting bored counting 1, 2, 3, …, 15 four times every night. I started to mix it up.

- I counted 1‑15 for the left side, then 16‑30 for the right, 31‑45 for the front, and 46‑60 for all around. But that got boring rather quickly, too.
- I switched to counting respectively by 1, 2, 3, and 4; that is, I’d count 1‑15 for the left, 17‑45 (by 2’s) for the right, 48‑90 (by 3’s) for the front, and 94‑150 (by 4’s) for all around. That was unsettling, though, because the boys started to think that 94, 98, 102, …, 150 were multiples of 4. While they recognized that counting by 2’s from 17‑45 gave the odd numbers, they weren’t able to discern that 94‑150 by 4’s analogously gave numbers
*n*≡ 2 mod 4. Nor did I expect them to — they’re only 3½. - I therefore switched it up again and counted by 1’s, 2’s, 3’s, and 6’s, which meant that “all around” was now 96, 102, 108, …, 150, which provided the more satisfying pattern of multiples of 6.

Then one night, Eli shocked me. He said, “Daddy, I figured out another way to count by 5’s.” I wasn’t really sure what he meant, but I assumed that “counting by 5’s” was a general term he was using to refer to skip counting. I asked him to explain. He said, “Like this: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55.”

Upon hearing this, I thought what any geeky American father would think. “Holy sh*t. Did my three-year-old son really figure out the triangular numbers by himself?”

So I asked him, “What are those numbers, Eli?”

“Well,” he began, “there’s 1,” and then 1 + 2 = 3, and 1 + 2 + 3 = 6, and 1 + 2 + 3 + 4 = 10, and…

Through a few more questions, I realized that he had ascertained these sums using my old TI‑83 calculator. I am proud to share a picture of the boys happily playing with technology:

The boys had been playing with some of my graphing calculators because they liked typing letters to spell words. Little did I know they were entering expressions and learning some math, too. (Take that, all you stodgy opponents of calculators in the classroom!)

After that, there were no holds barred; during teeth brushing, I started busting out all kinds of sequences:

- Triangular numbers: 1, 3, 6, 10, …, where
*T*(*n*) = ½(*n*)*(*n + 1). - Square numbers: 1, 4, 9, 16, …, where
*S*(*n*) =*n*^{2}. - Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, …, where
*F*(*n*) = F(*n*– 1) +*F*(*n*– 2). - Feeby numbers: 11, 22, 33, 44, 55, …, where
*Fb*(*n*) = 11*n*.

(These are just the multiples of 11, but Eli named them the Feeby numbers, I think because it sounded a little like the first part of Fibonacci numbers, and two of the numbers in the list (55, 66) are elements of the Fibonacci sequence.) - Perrin sequence: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, …, where
*P*(*n*) =*P*(*n*– 2) +*P*(*n*– 3).

I’ve been thinking about sharing Moser’s Circle Problem with them and showing them the sequence 1, 2, 4, 8, 16, 31, 57, 99, …. Then again, maybe not.

Okay, so if you’re one of the folks who came for a math joke and tolerated that entire story, then by gosh you deserve a math joke. I don’t have a joke about triangular numbers (note to self: create a joke about triangular numbers), but I do have a joke that involves the word *triangular.*

What is small, green, and triangular?

A small, green triangle.

Yeah, I know… it’s lame.

Let me try to make amends with a cool triangular number problem.

Append the digit 1 to the end of every triangular number. For instance, from 3 you’d get 31, and from 666 you’d get 6,661. Now take a look at all of the divisors of the numbers you’ve created. What are the units digits of the divisors for every number created in this way? Can you prove that this result always holds?

I have a proof, but you’ll have more fun solving it on your own than reading my solution.

### Pattern Recognition

My kids are three years old — yes, both of them are three years old; they’re twins — and they already know how to read and spell. I’ve always enjoyed watching kids’ intellectual development, but it’s all the more exciting when they’re your own children.

They’re pretty good with numbers, puzzles, and patterns, too, but yesterday they reminded me that they’re still just three years old. Alex, Eli, and our golden retriever Remy were riding in the car on the way to a local park. From the backseat, Eli asked, “How do you spell *park*, daddy?”

This is a word that Eli knows how to spell, so I said, “I don’t know. Can you help me spell it? What’s the first letter?”

That was all the prompting he needed. He spelled it easily: “P-A-R-K.”

“Great!” I said. “Let’s spell some other words. What’s the opposite of *light*?”

Alex said excitedly , “Dark!” and then spelled it correctly: “D-A-R-K.”

“Excellent!” I said. “And you have a cousin…”

“Mark!” Eli screamed. “M-A-R-K!”

“Hmm… *park*, *dark*, *Mark*… and what would Remy do if he saw a cat?” I asked, thinking that they’d see the pattern of rhyming words.

“He’d chase it,” Alex said.

I chuckled. “Well, yeah, he probably would. But what else might he do?”

“He’d eat it!”

Like I said, they’re only three. “I suppose he might,” I said.