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 3: Comparing Games

The topics for this third week is Comparing games. Students will determine the outcome of simple sums of games using inequalities.

- Dr. Tom MorleyProfessor

School of Mathematics

So, here's an outline of, of how to show that 1 over 2 to the n, minus up is

Â positive. Let's just look at the left going first

Â move. If left moves first, I'd claim that the

Â best move for left is to play to this star.

Â What's left is then 1 over 2 to the n plus star.

Â Now, now it's right's turn, what can right do?

Â Right can, can move from star to zero. Remember that the two, the only option in

Â star is for left to move to zero or for right to move to zero.

Â In which case, what's left is 1 over 2 to the n, left just chops the blue edge, and

Â now there's nothing left, and so left wins.

Â The other possibility is for right to cut off one of these red edges, and that just

Â delays things. So, right cuts off one of those edge,

Â edges the left then moves from star to 0, and what's left is right cuts off another

Â red edge, blue cuts off the blue edge, and now left wins.

Â So, so that's the argument when left goes first.

Â When left goes second, there's a little bit more stuff to do, but the argument is

Â very similar. And again, if left goes second, left wins.

Â Put this all together and that says that 1 over 2 to the n minus star is positive.

Â Which says, with what we did previously, that up is bigger than zero and less than

Â 1 over 2 to the n for any n. And there's no number that does that.

Â So, up is not a number. Okay.

Â That ends week 3. I'll post a little quiz and I'll, I'll

Â talk through it for the last part of week 3.

Â Okay. So, the quiz for week 3 is, I have a whole

Â bunch of games. You have to decide in each game whether

Â it's positive, negative, equal to zero or fuzzy with zero.

Â It can't be two of these, it has to be exactly one of these.

Â The first game is these hack and bush with three stems like this.

Â And second one, hack and bush with, with one green stem and one blue stem.

Â Game number three is 2 copies of one green stem and one red stem.

Â And number four is a little more abstract, go back and look at the definitions of up

Â and star and figure out whether up plus star is positive, negative, zero or fuzzy

Â with zero. Work this out and we'll post the solutions

Â by the time you see this. Take care.

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