Monday, July 14, 2014

Chains (part 1)

What: starting to investigate chains (description below)
Who: J1
When: mid-day, after breakfast
Where: on the floor in the middle of everyone
How: pencil and paper
Why: pattern seeking, exploration, thinking about extensions

To give you an idea of the challenge I presented to J1 today, we sat down and I told him that I was going to write down a sequence of numbers following a rule.  To start, I needed him to give me two numbers. He gave me 9 and 8.  I started writing the following, periodically asking him if he thought he knew the next number:

9 8 7 5 2 7 9 6 5 1
6 7 3 0 3 3 6 9 5 4
9 3 2 5 7 2 9 1 0 1
1 2 3 5 8 3 1 4 5 9
4 3 7 0 7 7 4 1 5 6
1 7 8 5 3 8 1 9 0 9 9 8

By that point, he had made many guesses, but hadn't gotten the rule.  I said that the sequence would repeat now and we talked about that a bit.  Basically, said it made sense because we had started with 9 and 8, so if that came up again it should be like starting again.

So, what's my rule?

Well, the point of this blog isn't particularly to set you math challenges (though I think this little exploration will still contain a lot of that). Here's what's going on and some of the things you can investigate.

A week ago, I saw a little warm-up exercise in RightStart Math called chains that caught my attention. They outline the following procedure:
1. pick one single digit numbers and write it down, for example 2.
2. pick another single digit number and write it down, for example 6.
3. Add them together and write down the ones-digit of the result.  Here, we would get 8.
4. Keep adding the last two numbers in your sequence and write down the ones digit of the result.
5. Continue until you repeat.
Staying with our example, our sequence would continue: 2 6 8 4 2 6.

You got the step 8+6 = 14, but only keep the 4, right? In case you think it this whole set-up was unfair to J1, just know that this type of modified addition was introduced at school, maybe even in kindergarten.

If we reversed the starting order, the sequence changes: 6 2 8 0 8 8 6 4 0 4 4 8 2 0 2 2 4 6 0 6 6 2

I've highlighted the two digits where the sequence repeats, but I'd actually prefer to leave those out and think of it as a 4 step chain and a 20 step chain.  The one J1 seeded for us was a . . . 60 step chain (isn't it helpful how I arranged the numbers so nicely in a little array?)

Beyond the basic mystery of guessing the rule (which you could leave out), what's interesting here?  Some things to consider that will get you started:

• Does the order of your starting digits matter? (we've already seen above that it does)
• will every starting pair of numbers eventually repeat (form a chain)?
• Will you ever get a repeat somewhere in the middle of your sequence, so the sequence looks something like [x] [ y] [y][y].... where [x] is one sequence of digits and [y] is another?
• How many chains are there?
• What length are all the chains?
• Will every chain contain all the digits 0 to 9?
Some little hints:  at any step, how much information do you need to continue the sequence? Can you reverse the sequence?

Further exploration
Dropping the tens digit actually means we are working modulo 10.  It really just means we are treating 10 the same as 0, but that simple idea is very powerful. We could have chosen a different number than 10, say 13 or 140 or 5 or 2 (though 1 is a bit of a problem). So . . . how do our chains look when you play with those cases? At the very least, we would start by asking the same questions we had before:

• Will every starting pair of numbers eventually repeat (form a chain)?
• Will you ever get a repeat somewhere in the middle of your sequence, so the sequence looks something like [x] [ y] [y][y].... where [x] is one sequence of digits and [y] is another?
• How many chains are there?
• What length are all the chains?
• Will every chain contain all the numbers 0 to n-1 (where we are working modulo n)?
I don't know if I'll get to this level with J1 for a while, but it is something I get to play with myself when I'm waiting at their swimming lessons.

1 comment:

1. Some notes for further investigation: Fib in Z_n
and
more Fib in Z_n

Both are from math.stackexchange.com