## Posts tagged ‘pattern’

### A Great Day for a Pattern

When I first saw today’s date in mm/dd/yy format — 11/12/21 — I thought, “Well, that’s pretty cool. It’s all 1s and 2s.” And then I thought, as I’m sure you did, “In ternary, that’d be 376,” because everyone thinks in ternary, right?

But then I looked at the number again, and I thought, “Ah, hello, old friend. Good to see you again.”

Those six digits form the fifth term of a famous pattern:

1

11

21

1211

111221

So your first question is, what’s the next term?

If you’ve never seen this pattern before, it’s worth a little of your time to try to figure it out before reading more about it at MathWorld.

Your second question — if you’re still reading — is, what’s the greatest digit that will ever appear in this sequence? As you can see above, the first five terms only contain 1s and 2s. What digits are in the sixth term? What digits will appear beyond the sixth term? How do you know?

### A Pattern Puzzle for the New Year

Over at the Visual Patterns site, the directions state that if you click on a pattern, you’ll get to see the number of objects in the 43rd step. Why 43? I assumed that it had to do with Fawn Nguyen being a fan of Troy Polamalu — which, as far as I’m concerned, would be just one more reason to have an infinite amount of respect for her — but when I asked about it, Fawn explained that 43 was chosen as…

…a random number that was farther down the step number to prevent students from finding the number of objects recursively, but not too far.

This explanation sits well with my beliefs. In my book *One-Hundred Problems Involving the Number 100*, I stated that it’s appropriate to ask students to find the 100th term in a sequence because 100 is “big enough to exhilarate, but not so big as to intimidate.” The same could be said about 43.

Following Fawn’s lead, here’s a problem to get you in the spirit for the new year. Feel free to share this problem with your students on or near January 1.

*How many squares would be in the 43rd element of this sequence?*

Coincidentally, I shared this sequence with Fawn, and it now appears as #392 on the Visual Patterns site.

Speaking of sequences, here’s my favorite infinite sequence joke.

Infinitely many mathematicians walk into a bar. The first says, “I’ll have a beer.” The second says, “I’ll have half a beer.” The third says, “I’ll have a quarter of a beer.” They continue like this, each one ordering half as much as the last. The barman stops them and pours two beers. One of the mathematicians says, “That’s it? That’s not enough for all of us!” The bartender replies, “C’mon, folks. Know your limits.”

For fun, figure out how much beer the 43rd mathematician asked for.

And as a little more fun, guess the value of all the coins in the glass below. As a hint, there are the same number of quarters, dimes, and nickels, but three times as many pennies as dimes. (Said another way, Q:D:N:P::1:1:1:3.)

If you think about it a little, you’ll realize the answer without doing any computation.

Happy New Year!

### Stick Figure Math

I’ll never forget the first time I saw the pattern

1, 2, 4, 8, 16, __

and was dumbfounded to learn that the missing value was **31**, *not 32*, because the pattern was *not* meant to represent the powers of 2, but rather, the number of pieces into which a circle is divided if *n* points on its circumference are joined by chords. Known as Moser’s circle problem, it represents the inherent danger in making assumptions from a limited set of data.

Last night, my sons told me about the following problem, which they encountered on a recent math competition:

*What number should replace the question mark?*

Well, what say you? What number do you think should appear in the middle stick figure’s head?

Hold on, let me give you a hint. This problem appeared on a multiple-choice test, and these were the answer choices:

- 3
- 6
- 9
- 12

Now that you know one of those four numbers is *supposed* to be correct, does that change your answer? If you thought about it in the same way that the test designers intended it, then seeing the choices probably didn’t change your answer. But if you didn’t think about it that way and you put a little more effort into it, and you came up with something a bit more complicated — like I did — well, then, the answer choices may have thrown you for a loop, too, and made you slap your head and say, “WTF?”

For me, it was Moser’s circle problem all over again.

So, here’s where I need your help: **I’d like to identify various patterns that could make any of those answers seem reasonable.**

In addition, I’d also love to find a few other patterns that could make some answers other than the four given choices seem reasonable.

For instance, if the numbers in the limbs are *a*, *b*, *c*, and *d*, like this…

then the formula 8*a* – 4*d* gives 8 for the first and third figures’ heads and yields 8 × 6 – 4 × 9 = **12** as the answer, which happens to be one of the four answer choices.

Oh, wait… you’d don’t like that I didn’t use all four variables? Okay, that’s fair. So how about this instead: ‑3*a* + *b* + *c* + 2*d*, which also gives ‑3 × 6 + 7 + 5 + 2 × 9 = **12**.

Willing to help? **Post your pattern(s) in the comments.**

[**UPDATE (3/9/18):** I sent a note to the contest organizers about this problem, and I got the following response this afternoon: “Thanks for your overall evaluation comments on [our] problems, and specifically for your input on the Stick Figure Problem. After careful consideration, we decided to give credit to every student for this question. Therefore, scores will be adjusted automatically.”]

### 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.