Discrete/n by m prob

Anyone have any idea how to do #3 on:

http://www.bioalgorithms.info/moodle...ks/hw0/hw0.pdf

My professor said she solved it using a checkerboard.

Thanks.

now if it weren't password protected that would be nice.

Shouldn't be. I certainly did not need a password.

http://www.bioalgorithms.info/moodle...ks/hw0/hw0.pdf

Hmm. Not sure why you're getting a password. I can type it out if need be.

Quote:

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Originally Posted by Ideasman

Shouldn't be. I certainly did not need a password.

http://www.bioalgorithms.info/moodle...ks/hw0/hw0.pdf

Hmm. Not sure why you're getting a password. I can type it out if need be.

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Because you probably have a cookie set, but you won't get much response
to your question if we have to create an account and or login to see it.

RonL

Quote:

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Originally Posted by Ideasman

I can type it out if need be.

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Please do.

Two players play the following game with two “chromosomes” of length
n and m nucleotides. At every turn a player can destroy one of the chromosomes and break another one into two nonempty parts. For example, the first player can destroy a chromosome of length n and break another chromosome into two chromosomes of length m/3 and m − m/3 . The player left with two single-nucleotide chromosomes loses.

Who will win? Describe the winning strategy for each n and m.

I still have no idea where to start with this problem. For any n and m, I am assuming there might be 2 different cases for even/odd?

Quote:

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Originally Posted by Ideasman

Two players play the following game with two “chromosomes” of length
n and m nucleotides. At every turn a player can destroy one of the chromosomes and break another one into two nonempty parts. For example, the first player can destroy a chromosome of length n and break another chromosome into two chromosomes of length m/3 and m − m/3 . The player left with two single-nucleotide chromosomes loses.

Who will win? Describe the winning strategy for each n and m.

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Quote:

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Originally Posted by Ideasman

I still have no idea where to start with this problem. For any n and m, I am assuming there might be 2 different cases for even/odd?

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Solve the game backwards as in this thread and this one.

Um,

LWLWLWLW
WLWLWLWL
...
...
...

Don't the L's and W's just alternate?

EDIT: Err, nevermind, every other row is W's?

I spent some more time looking at the problem. I tried doing a 13 by 13 matrix.

I got the following:

Code:

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LWLWLWLWLWLWL
WWWWWWWWWWWWW
LWLWLWLWLWLWL
WWWWWWWWWWWWW
LWLWLWLWLWLWL
WWWWWWWWWWWWW
LWLWLWLWLWLWL

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...and the pattern just repeats.

Assuming this is right, I think the strategy would be to give your opponent a pile of odd # nucleotides and another pile of odd nucleotides; if you give your opponent a pile of even and a pile of odd, they will win (assuming they know the strategy). Player one will always win if he goes first (will win if at least one chromosome pile has an even number of nucleotides). Player 2 will always win if the start piles both have an odd # of nucleotides.

does it have to be m/3, or could it be m/4 or anything the player chooses.

Has to be m/3, or at least that's how I interpret the problem.

Quote:

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Originally Posted by Ideasman

Two players play the following game with two “chromosomes” of length
n and m nucleotides. At every turn a player can destroy one of the chromosomes and break another one into two nonempty parts. For example, the first player can destroy a chromosome of length n and break another chromosome into two chromosomes of length m/3 and m − m/3 . The player left with two single-nucleotide chromosomes loses.

Who will win? Describe the winning strategy for each n and m.

---

Quote:

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Originally Posted by Quick

does it have to be m/3, or could it be m/4 or anything the player chooses.

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Quote:

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Originally Posted by Ideasman

Has to be m/3, or at least that's how I interpret the problem.

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It looks like that is just an example to me. The rule stated was:

At every turn a player can destroy one of the chromosomes and break another one into two nonempty parts.

Then came the example with m/3.

Meh, hard problem. Then I have no clue. And I thought I had it, too :( .

Quote:

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Originally Posted by Ideasman

I spent some more time looking at the problem. I tried doing a 13 by 13 matrix.

I got the following:

Code:

---

LWLWLWLWLWLWL
WWWWWWWWWWWWW
LWLWLWLWLWLWL
WWWWWWWWWWWWW
LWLWLWLWLWLWL
WWWWWWWWWWWWW
LWLWLWLWLWLWL

---

...and the pattern just repeats.

Assuming this is right, I think the strategy would be to give your opponent a pile of odd # nucleotides and another pile of odd nucleotides; if you give your opponent a pile of even and a pile of odd, they will win (assuming they know the strategy). Player one will always win if he goes first (will win if at least one chromosome pile has an even number of nucleotides). Player 2 will always win if the start piles both have an odd # of nucleotides.

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Quote:

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Originally Posted by Quick

does it have to be m/3, or could it be m/4 or anything the player chooses.

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Quote:

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Originally Posted by Ideasman

Has to be m/3, or at least that's how I interpret the problem.

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Quote:

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Originally Posted by JakeD

It looks like that is just an example to me. The rule stated was:

At every turn a player can destroy one of the chromosomes and break another one into two nonempty parts.

Then came the example with m/3.

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Quote:

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Originally Posted by Ideasman

Meh, hard problem. Then I have no clue. And I thought I had it, too :( .

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I think your solution is correct as is. The m/3 restriction is not needed.

A simple proof by induction is possible. Establish the base case that a 2-pile is a winning position and a 3-pile is a losing one. Then for n > 3, an n-pile with n even is winning because it breaks into two odd piles, both of which are losing for the opponent. An n-pile with n odd is losing because it must break into an even pile and an odd pile. The even pile is a winning one for the opponent.