This course will cover the mathematical theory and analysis of simple games without chance moves.

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From the course by Georgia Institute of Technology

Games without Chance: Combinatorial Game Theory

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This course will cover the mathematical theory and analysis of simple games without chance moves.

From the lesson

Week 4: Numbers and Games

The topics for this fourth week is Simplicity and numbers. How to play win numbers. Students will be able to determine which games are numbers and if so what numbers they are.

- Dr. Tom MorleyProfessor

School of Mathematics

>> Games without chance. Okay.

Â I just dealt myself a card hand by chance. And now look at what I got.

Â You believe that? Okay.

Â But, not a chance mix. This is the module.

Â Simplicity. It's all about simplicity.

Â Okay, let's look. What do I mean by simplicity?

Â Remember last time we looked at things like, well let's do minus 3, 1.

Â Now, it turns out that this is some number, and the number has to be between 1

Â and minus 3. Whenever you have a game like this that's

Â a number, all the options over here are less than, strictly less than all the

Â options over here. It turns out that unless that happens,

Â that's not a number. There are games, lots of games that aren't

Â numbers. But if it is a number, it has to be

Â somewhere in between. But, you see we have lots of choices here,

Â you know, minus 2 is in between, minus 1.12345 is in between.

Â All kinds of numbers are in between, and if you actually believe in real numbers,

Â which some people do. I happen to believe in real numbers

Â sometimes. There are infinitly many numbers in

Â between, and so, we have to decide, which one.

Â And the answer is, the simplest one. Simplest one.

Â So, This is some number. It's got to be, if this is some number,

Â it's got to be, be between minus 3 and 1, and the question is which one.

Â And the answer is the simplest number. Now, 0 is the simplest number of all.

Â And then you have 1, 2, 3, and also their negatives.

Â And then slightly more complicated than these are 1 half minus a half, numbers

Â that are, 3 and a half would be a, actually, more complicated than 1 half,

Â because it's bigger. And then is the quarters, and then the

Â 8ths, and then the 16ths, etc., etc. Okay.

Â So we'd look at the game 1, and 1 and 3/8. And ask, what's the simplest number that's

Â in between? Well, it turns out there's a number where

Â the 4th. Namely 1 and a quarter that's in between

Â these 2. Even though this is.

Â Involves 8ths. We don't have to go to 8th or 16ths to fit

Â something in between. We can actually get something strictly in

Â between. That's got a, that's got a 4 in the

Â denominator. And having a 4 in the denominator is

Â simpler than having an 8th in the denominator.

Â So, so this is, 1 and a quarter which is in between 1 and 1 and 3 8th.

Â So, when there's a choice Of, of which one to fit in between is the simplest one.

Â Now let me actually explain something. One more thing.

Â Take a look at say one. And 3 and 7 8ths.

Â Okay, there's lots of numbers that fit in between here and the, The value here X

Â that is, this is equal to a number X. The value of X is not the average of these

Â two. It's the simplest that's fixed in between

Â There's an actually a whole number that fits in between and so it's two.

Â Okay, you can try one. You can try this at home.

Â Let's try 3 8th, 3 quarters. What's the simplest number that's bigger

Â than 3 8ths and less than 3 4ths? Try it at home.

Â I think you'll find that, that number is halfway in between 0 and 1.

Â Just to give away the answer. So, and to prove that this number is equal

Â to 1 half, you would have to show this number, this game, minus this game, minus

Â 1 half, is a 0 game, that is, a loss for the first player.

Â So there we have it.

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