Math Modeling - Dynamical Systems -

[purpleehobbit]

***Ugh, I'm so sorry. I think I put this in the wrong thread. It probably should be in Calculus and Beyond... I tried to delete it but didn't see that option.So my professor gave us this problem to work on but no one could figure it out. I don't know where to even start and there's nothing at all similar in our book that I can find. It's driving me crazy because he said it was the most basic kind of modeling problem. And then he made me feel dumb...again...when he commented on my not being able to figure it out. (This is my 3rd class with him, I did well in his first class so he expects the same... I've just been a bit slower at figuring out this class)

Homework Statement

So he told us about this game between two players. Player 1 chooses any number 1 - 10, including 1 and 10. Player 2 takes that number and adds any number 1 - 10 including 1 and 10. The two players take turning adding a number not greater than 10 to the previous number until one player reaches 100 and that player wins.

Then he said to figure out a strategy for winning.

It was the first lecture so I'm assuming it's on what he covered which was dynamical systems and difference equations.

Homework Equations

These are the equations he showed us:

First difference equation

$$\Delta$$a_n = a_n+1 - a_n

Equilibrium value

a = b / (1-r)

Solution to a dynamical system

a_n = crⁿ = b / (1-r)

b, c, and r are constants.

The Attempt at a Solution

I think he wants us to create a dynamical system with difference equations and find a solution. I know how to do that if we're talking about mortgages, annuities, interest rates, payments, etc.

I have no idea how to do this kind of system with this game.

What I do know is that the final result needs to be 100. In order to win, you need to be the one that gets to add to 89. So you wouldn't want to use 79.

I'm not asking for anyone to solve it for me. Just maybe some ideas on where to start. I'd settle for a link to a website that describes this kind of modeling.

Thanks in advance for any help you might offer.

 

[Slats18]

Hi purpleehobbit =)

Don't know much about dynamical systems and how to model them, but I thought I throw something in, just in case it helps haha.

The method to win here is to be the player that first reaches the number 11n+1 for any natural number up to 9 (obviously haha), and then keep the pattern. You're right in saying that 89 is the number to get to, n=8, but the lowest possibility is 1. If you go first, choose 1, you should win as no matter what number player two takes, you can reach the next magic number.

Ex: You take 1, player two takes 5. Then, you take 6, to reach 12, a solution to 11n+1.

This will work every single time, IF you go first. If you go second, then 9 times out of 10 on the first choice, you will win also, as the only time you can lose is if the other player takes 1 and knows the method.


So there's a strategy for winning, but not in dynamical terms, I apologise for that, but I hope this helps =)