Sideways arithmetic

Lately Archer has been interested in creating word addition problems of the type he reads in his beloved More Sideways Arithmetic from Wayside School.  Wikipedia tells me that these problems are called "cryptarithms"; an example would be "one plus one equals zero."  The puzzle is to find out what digit each letter represents.

Archer has been trying to make his own, but the trick of constructing them so that all the elements are real words is slightly beyond him.  (He did offer "no plus no equals too" today.)  Yesterday he emerged from his room with "problems plus problems equals srbenpolm," and suggested -- without prompting -- that "srbenpolm" might be a city in a foreign country.

Cady Gray tried her hand at one last night.  She brought me out a sheet last night headlined "Bonus."  On it she had constructed "quail plus quail equals lequqi."  As a starting point she gave me that E=3 and L=1.  Can you figure out what Q, U, A, and I are?

Tonight, tonight the highway's bright

Today's post about otherworldly geometries realized in crochet and cotton is at Toxophily.

And if you want to read more about hyperbolic space and the way mathematicians figured out how to model it with crochet, here's a good place to start.

These are a few of my favorite things

At church yesterday, during the peace, Archer was cornering our unsuspecting pewmates and telling them: "Now, my favorite way to count is like this: 1, 1, 2, 3, 5, 8, 13, 21, 34 ..."

So the note I tucked into his lunch today asked this question: "Why do you love the Fibonacci sequence?"

Here are his answers (I provided the numbered blanks):

1. It is calculated 1+1=2, 1+2=3 [AA provided arrows showing how each number repeats in the next equation]
1. It starts 1, 1, 2, 3, 5 [AA circled the bolded numbers]
2. It has a puzzling ctdn.
3. The 12th number is 144 (12x12!)
5. The countdown is hidden in this note!

And along the edges he provided the first 12 numbers in the sequence.

Tomorrow it's Crazy Hat Day for the drug-free emphasis Red Ribbon Week in school, and I'll put the kids in their fish hats. Archer could care less about having a fish on his head, but he might get a kick out of the fact that the stripes are all Fibonacci numbers. Now there's something he wouldn't mind dressing in -- or living in.

Combinations

A couple of days ago, Cady Gray brought home a piece of kindergarten "homework." It was a game where we took turns pulling two cut-out pictures out of a paper bag; if we got a pair of pictures that rhymed, we got to keep the match.

Archer was really interested in this game, and kept playing long after Cady Gray had moved on to something else. I idly asked him how many different combinations of two cards we could draw from the sixteen cards in the bag. "If it mattered which order they were in, the answer would be sixteen factorial," I said, "but I don't know how to figure out the answer when the order doesn't matter."

That night I was packing lunches for the kids, and as always, I wrote notes to accompany them. Since this summer I've been writing little quizzes or questions for Archer, and packing a pencil in the box so he can answer them. Because I was still wondering what he would do to figure out the answer, I wrote: "If there are eight pairs of rhyming cards in the sack, and the order doesn't matter, what are the chances you will pull out matching cards? Total # of possible combos: ___ Chance of a match: ____"

When I got home yesterday afternoon, I immediately opened his lunchbox to see what he'd written. In the first blank, he'd penciled in "120." In the second, "8 in 120."

Now I had to try to figure out what the right answer was. I Googled around until I found a description of how to figure out problems with the form "16 choose 2" using Pascal's triangle. Then I Googled around some more until I found an image of Pascal's triangle with enough rows to get down to 16. I read over to the second place and ...

120. He was right.

But how did he know? I asked him today, and he explained that he added 1+2+3+4 and so on, up to 15. When I asked him how he knew how to do this, he just said that if he kept going farther, he would draw one of the combinations that had already been used. Noel asked him if he read about how to do this in a book, and he said that he hadn't.

I'm at a loss to understand what numerical logic would lead him to this method. Any mathematicians out there want to explain it to me in plain English?

Sets and equations

I just finished reading a book called Is God A Mathematician? by Mario Livio. (I'll be reviewing it for the A.V. Club soon.) It's an entertaining historical guide to the debate between formalists -- those who believe mathematics is a human construct that we apply to the world -- and realists -- those who consider math the inherent language of nature itself.

It's hard not to think of Archer while exploring both sides of this debate. This morning Archer was talking about words that don't count in Scrabble, and Noel commented that some proper nouns are also words in the dictionary -- like "archer." Archer immediately jumped in: "Yeah, and Archer scores 1 -- 2 -- 5 -- 9 -- 11 points." It's a source of endless delight to him that almost everything he encounters can be associated with a number or equation.

I asked Archer this past week, after a similar conversation, whether it made him happy that the world comes equipped with numbers, and he assented with his usual, "Oh, yeah." In some ways I thank my lucky stars that Archer is growing up in a world where so many numbers are already assigned to its features -- highways, temperatures, addresses, library books, schedules, sports. Time and space are mapped onto Cartesian coordinates, and everything you consider can be delineated by an integer, and those integers can be manipulated and explored in innumerable ways.

In another time or place, he might have had to assign his own numbers to the elements of his environment. Or maybe there would not have been enough enumeration in his culture to allow his fixation on mathematics to bloom and grow. Whether the numerical nature of the world is discovered or invented, it exists for him, and it's allowed him to be who he is. And I wouldn't have it any other way.

To infinity and beyond

Today's post with springtime handknit socks repurposed for fall is at Toxophily.

Archer wrote a poem on a sticky note and left it on our front picture window this morning. He's fascinated by the idea that integers ("the counting numbers," as it's put in his schoolbooks) go on and on forever.

The 123s

I'm 1, 2, 3, you know.
I will not have 0.
I will have no stop!

Can I get into a more exclusive club?

Archer ginned up this computer printout for me yesterday -- a reproduction of the records he gets when he takes a reading comprehension test at school. My reading skills need some work, apparently, even though I've read that book approximately three dozen times in the past five years. I don't think I deserve to be in the 12.5% club.

From blog elements

Meanwhile, Cady Gray asked for a sheet of addition problems to do before leaving for school today. (I wrote the problems, she wrote the answers, and I tried to give her 100% with a red pen that wouldn't make a mark.) And between last weekend and today, she's figured out the concept of division. Lest we think that she's just a smaller version of her number-cruncher brother, she also delights in telling us that she has a boyfriend at preschool named Sam. "Sam says, 'I'll hug her but I won't kiss her!'" she repeats amid gales of laughter.

From blog elements

My kids are pretty awesome

I hate to brag, but my kids are pretty awesome. Here's the proof.

Exhibit A: A rough transcript of Cady Gray, age barely 4, solving the addition problem 106+355 in her head about five minutes ago.

"Ohhh-kay. So. 6+5=11, so I put a 1 in the tens place, and I put down a 1. 1+0+5, so 1+0=1, and then the answer is 6. Now. 2+3 ..."

"The problem was 106+355."

"Mom, do not give me three digits. Ohhh-kay. So 1+3=4. So the answer is ... four ... hundred .... sssssssssixty ... one. 461!"

Exhibit B: An acrostic poem Archer wrote in school -- age 7 years 1 month, second grade, Arkansas, USA, the universe.

F alling leaves
A ll falling from the trees to the
L and.
L et's let fall go away.
..........Whoosh!
............ And
............... now
.................. it's
..................... gone.

Consumed

Ravelympics continues to devour my free time, and there's an update (on the actual knitting, for a change) in today's post over at Toxophily.

Meanwhile, Cady Gray has decided to become a math genius like her brother. She demands multiplication problems from us, then laboriously (but with astounding accuracy) figures them out with a combination of counting on her fingers, skip counting and adding when she gets past her memorization zone, and somehow remembering where she is the whole time.

Tonight at dinner she boasted that she could add 24+4 (a relatively straightforward one). "28!" she announced. "How do you know that?" I demanded. "My brain is connecting to my smarties," she explained. Then, in a dramatic whisper: "It's connecting right now!"