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) = n².
• 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) = 11n.
  (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.

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