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.

Printable View

- October 29th 2006, 11:19 AMIdeasmanDiscrete/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. - October 29th 2006, 12:23 PMQuick
now if it weren't password protected that would be nice.

- October 29th 2006, 12:38 PMIdeasman
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. - October 29th 2006, 12:47 PMCaptainBlack
- October 29th 2006, 12:53 PMQuick
- October 29th 2006, 01:22 PMIdeasman
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. - November 1st 2006, 11:57 PMIdeasman
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?

- November 2nd 2006, 01:02 AMJakeD
Solve the game backwards as in this thread and this one.

- November 2nd 2006, 02:59 PMIdeasman
Um,

LWLWLWLW

WLWLWLWL

...

...

...

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

EDIT: Err, nevermind, every other row is W's? - November 2nd 2006, 04:51 PMIdeasman
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

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. - November 2nd 2006, 05:14 PMQuick
does it have to be m/3, or could it be m/4 or anything the player chooses.

- November 2nd 2006, 06:08 PMIdeasman
Has to be m/3, or at least that's how I interpret the problem.

- November 2nd 2006, 08:15 PMJakeD
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. - November 2nd 2006, 08:18 PMIdeasman
Meh, hard problem. Then I have no clue. And I thought I had it, too :( .

- November 3rd 2006, 01:45 AMJakeD
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.