Putnam 2008/A2

Mathematical Competitions
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Demetres
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Putnam 2008/A2

#1

Post by Demetres »

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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Tolaso J Kos
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Re: Putnam 2008/A2

#2

Post by Tolaso J Kos »

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.
Demetres
Former Team Member
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Re: Putnam 2008/A2

#3

Post by Demetres »

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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