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 Post subject: Putnam 2008/A2Posted: Mon Nov 09, 2015 3:02 pm

Joined: Mon Nov 09, 2015 11:52 am
Posts: 77
Location: Limassol/Pyla Cyprus
Alan and Barbara play a game in which they take turns filling entries of an initially empty $2008 \times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all the entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?

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 Post subject: Re: Putnam 2008/A2Posted: Wed Nov 11, 2015 2:55 pm

Joined: Sat Nov 07, 2015 6:12 pm
Posts: 841
Location: Larisa
Demetres wrote:
Alan and Barbara play a game in which they take turns filling entries of an initially empty $2008 \times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all the entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?

Hi Demetres,

I vote for Barbara, since she can use the following strategy:

Whenever Alan writes a number $x$ in an entry in some row, Barbara writes $-x$ in some other entry in the same row. At the end, the resulting matrix will have all rows summing to zero, so it cannot have a full rank.

_________________
Imagination is much more important than knowledge.

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 Post subject: Re: Putnam 2008/A2Posted: Wed Nov 11, 2015 4:30 pm

Joined: Mon Nov 09, 2015 11:52 am
Posts: 77
Location: Limassol/Pyla Cyprus
Cool. Another strategy for Barbara is to play in such a way as to make the first two rows identical: Because there is an even number of entries outside the first two rows it is easy for Barbara to achieve so.

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