## Bragging

Last week my son and I read a kids' book to each other. We took turns in which he read five pages and I read two pages. He started the book, and I finished. There were 28 pages. How many turns reading did I have, and how many did my son have?

I was about to ask my son that question, when he asked me instead. I said "Get a piece of paper and a pencil." He did, and came back to the couch, saying "And Dad, don't help me."

I didn't. I didn't even watch. He scribbled for a while. I figured it out in my head, using a serious mathematical proof, plus algebra, that product of Middle Eastern Islamic scholarship. At first I thought it would be a two-variable problem, until I saw through that.

Then my son said the answer was that we each had four turns.

Exactly right.

My solution has two parts.

1. -- Equal number of turns.

Because he started and I finished, we had an equal number of alternating turns. If the number of turns was one each, the sequence would have looked like this, where S stands for a son turn and D stands for a Dad turn:

SD

No matter how many turns there were, If he started and I finished, the number of alternating turns would be equal, even here:

SDSDSDSDSDSDSDSDSDSDSDSDSDSD

So #S = #D.

I am sure that there is a mathematical name for this obvious principle. It may even be an axiom, but I don't know what it is. Help, anyone?

2. -- Algebra.

Let's call the number of turns T. Recall that #S = #D= T.

He read five pages per turn and I read two. There were 28 pages. So...

5T + 2T = 28.

7T = 28

T = 28/7

T = 4

I showed my son this solution, and I don't think he understood it. But I emphasized to him that his solution was perfect, and it had worked just right.

He explained his solution to me, and I understood that he had basically counted off the pages, making a mark on one piece of paper for him and one for me. That means that he kept a running total.

He gave me the piece of paper that showed how many I had read, and for two days I looked at it and thought that it was gobbledegook. It looked like he had scribbled numbers over numbers. Some numbers looked like letters. It made no sense. Then, yesterday, I figured it out.

If you refer to the image below, you'll see seven boxes, four with writing in them. What he did was write down the page numbers I had read. 6 and 7. Then 13 and 14. Then 20 and 21. Then 27 and 28.

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