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forum.virtualchallengemeets.com • View topic - Eugene Problem!(Game Theory) USAMO 2004 #4

Eugene Problem!(Game Theory) USAMO 2004 #4

Pose Olympiad-level questions to the monkeys in the peanut #.

Eugene Problem!(Game Theory) USAMO 2004 #4

Postby zefuri » Fri Apr 20, 2007 11:56 am

Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number in that row is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't. (If two squares share a vertex, Alice can draw a line from one to the other that stays in those two squares.) Find, with proof, a winning strategy for one of the players.
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Postby stupidityismygam » Fri Apr 20, 2007 7:57 pm

Bob kills Alice with a candlestick in the Library, and thus automatcally wins [unparseable or potentially dangerous latex formula]

EDit: forgot to mention the weapon/place
Last edited by stupidityismygam on Fri Apr 20, 2007 8:00 pm, edited 1 time in total.
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Postby zefuri » Fri Apr 20, 2007 7:59 pm

Your solution is not rigorous because you have failed to mention with what weapon Bob kills Alice, you just assumed that was trivial.
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Postby stupidityismygam » Fri Apr 20, 2007 8:06 pm

Alternate Solution:

WLOG assume the grid is on a chalkboard and there are many different colors of chalk (namely red,blue,white)

Bob throws Alice's red chalk in the air, and in an attempt to catch it she opens her mouth because she is awesome like that. Quickly, Bob chunks the blue and white chalk into the open mouth of Alice. Alice is stunned by this action of Bob and immediately spits out the chalk. Little does she know, Bob coated the chalk in poison and she instantly dies (after spitting out the chalk). Since she is dead, the red chalk falls on the ground and breaks into billions of pieces (you have to lemma that gravity works). Thus, the red chalk is no longer available for use (because Bob inhaled these pieces of chalk (no poison)). Thus, no red can be on the grid which is on the chalkboard. So, Bob wins. Alice dies. And everybody is happy (except of course the people who love and care for Alice). [unparseable or potentially dangerous latex formula]
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Postby stupidityismygam » Fri Apr 20, 2007 8:14 pm

WARNING SERIOUS QUESTION:

row and column, or just row?
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Postby zefuri » Fri Apr 20, 2007 8:50 pm

just row
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Postby makashi » Tue May 08, 2007 3:31 pm

Claim: If the black square is in square [unparseable or potentially dangerous latex formula], Bob can force the black square to be in square [unparseable or potentially dangerous latex formula] in an adjacent row.

Proof: Group the rows as follows.
Red group: Rows 1 and 2.
Blue group: Rows 3 and 4.
Black group: Rows 5 and 6.

Each time Alice adds a number to a row in column [unparseable or potentially dangerous latex formula], Bob should add a number in column [unparseable or potentially dangerous latex formula] to the other row in the group that is relatively the same compared to the other numbers in that row.

For example, if Alice adds the first number to row 3 column 1, then Bob simply needs to add any rational number to row 4 column 4 which is not already in the grid.

Suppose Alice adds the second number to row 5 in column 3 and that number happens to be larger than the other number in row 5. Then Bob needs to add a number to row 6 in column 6 which is larger than the other number in row 6. We know there is another number in row 6 and also that row 6 column 6 is open by symmetry of the moves.

Suppose Alice adds a number [unparseable or potentially dangerous latex formula] to a row, say row 1, column [unparseable or potentially dangerous latex formula] which already has at least two numbers in that row. Let the largest number in row 1 smaller than [unparseable or potentially dangerous latex formula] be in column [unparseable or potentially dangerous latex formula] and the smallest number larger than [unparseable or potentially dangerous latex formula] be in column [unparseable or potentially dangerous latex formula]. Then Bob shall write a rational number in row 2 column [unparseable or potentially dangerous latex formula] which is larger than the number in row 2 column [unparseable or potentially dangerous latex formula] and smaller than the number in row 2 column [unparseable or potentially dangerous latex formula]. Again, by symmetry, the square in which Bob needs to write his number will be unoccupied and the squares with which Bob needs to make comparisons will be occupied.

By symmetry of the construction, if the black square is in square [unparseable or potentially dangerous latex formula], Bob can force the black square to be in square [unparseable or potentially dangerous latex formula] in an adjacent row.

Therefore, the black squares in any group will not share a vertex under this strategy, and Bob wins.
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