Mixing
Infinite grid colouring game
$begingroup$
Alice and Bob have an infinite sheet of paper with a square grid on it. They play the following game: taking turns, they paint the sides of the squares (Alice is using the red colour, and Bob is using the blue colour); it is not allowed to paint any side which has been painted already. Alice was the first to make the move. Prove that Bob can play in such a way that Alice could never make a closed red contour.
I have been thinking about this question for several days, and I haven't found a way to answer it. I am assuming that there is an invariant/monovariant to exploit. The question was posed exactly as written above, on an undergraduate problem-solving sheet that I stumbled across online. Every other problem on the sheet could be solved quickly in a slick manner, so I'm presuming that this one can be too.
problem-solving
asked Jan 27 at 22:40
Cathal Ó CléirighCathal Ó Cléirigh
363
$endgroup$
add a comment |
$begingroup$
Alice and Bob have an infinite sheet of paper with a square grid on it. They play the following game: taking turns, they paint the sides of the squares (Alice is using the red colour, and Bob is using the blue colour); it is not allowed to paint any side which has been painted already. Alice was the first to make the move. Prove that Bob can play in such a way that Alice could never make a closed red contour.
I have been thinking about this question for several days, and I haven't found a way to answer it. I am assuming that there is an invariant/monovariant to exploit. The question was posed exactly as written above, on an undergraduate problem-solving sheet that I stumbled across online. Every other problem on the sheet could be solved quickly in a slick manner, so I'm presuming that this one can be too.
problem-solving
asked Jan 27 at 22:40
Cathal Ó CléirighCathal Ó Cléirigh
363
$endgroup$
add a comment |
$begingroup$
Alice and Bob have an infinite sheet of paper with a square grid on it. They play the following game: taking turns, they paint the sides of the squares (Alice is using the red colour, and Bob is using the blue colour); it is not allowed to paint any side which has been painted already. Alice was the first to make the move. Prove that Bob can play in such a way that Alice could never make a closed red contour.
I have been thinking about this question for several days, and I haven't found a way to answer it. I am assuming that there is an invariant/monovariant to exploit. The question was posed exactly as written above, on an undergraduate problem-solving sheet that I stumbled across online. Every other problem on the sheet could be solved quickly in a slick manner, so I'm presuming that this one can be too.
problem-solving
asked Jan 27 at 22:40
Cathal Ó CléirighCathal Ó Cléirigh
363
$endgroup$
Alice and Bob have an infinite sheet of paper with a square grid on it. They play the following game: taking turns, they paint the sides of the squares (Alice is using the red colour, and Bob is using the blue colour); it is not allowed to paint any side which has been painted already. Alice was the first to make the move. Prove that Bob can play in such a way that Alice could never make a closed red contour.
I have been thinking about this question for several days, and I haven't found a way to answer it. I am assuming that there is an invariant/monovariant to exploit. The question was posed exactly as written above, on an undergraduate problem-solving sheet that I stumbled across online. Every other problem on the sheet could be solved quickly in a slick manner, so I'm presuming that this one can be too.
problem-solving
problem-solving
asked Jan 27 at 22:40
Cathal Ó CléirighCathal Ó Cléirigh
363
asked Jan 27 at 22:40
Cathal Ó CléirighCathal Ó Cléirigh
363
asked Jan 27 at 22:40
Cathal Ó CléirighCathal Ó Cléirigh
363
asked Jan 27 at 22:40
Cathal Ó CléirighCathal Ó Cléirigh
363
asked Jan 27 at 22:40
Cathal Ó CléirighCathal Ó Cléirigh
363
363
add a comment |
add a comment |
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