Professor Halfbrain has spent his last weekend with analyzing $n\times n$ chessboards. Halfbrain says that a subset $S$ of squares on such a chessboard is queen-connected, if a chess queen can move around on $S$ and reach all the squares in $S$ while making only legal queen moves. (The queen is only allowed to move from squares in $S$ to squares in $S$, but the move may pass over other squares that are not necessarily in $S$. Every square in $S$ may be visited many times.)

Professor Halfbrain's first theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red, then these red squares form a queen-connected set.

 

Professor Halfbrain's second theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red and blue, then the red squares or the blue squares (or both) form a queen-connected set.

 

Professor Halfbrain's third theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red and blue and yellow, then the red squares or the blue squares or the yellow squares (or two of these sets, or all three sets) form a queen-connected set.

 

Professor Halfbrain's fourth theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red, blue, yellow and green, then at least one of the color classes must form a queen-connected set.

We all know Professor Halfbrain as an honorable and trustworthy mathematician, but even the best mathematician may sometimes be mistaken. Hence the question: Which of the Professor's four theorems do actually hold true?

Professor Halfbrain has spent his last weekend with analyzing $n\times n$ chessboards. Halfbrain says that a subset $S$ of squares on such a chessboard is queen-connected, if a chess queen can move around on $S$ and reach all the squares in $S$ while making only legal queen moves. (The queen is only allowed to move from squares in $S$ to squares in $S$, but the move may pass over other squares that are not necessarily in $S$. Every square in $S$ may be visited many times.)

Professor Halfbrain's first theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red, then these red squares form a queen-connected set.

Professor Halfbrain's second theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red and blue, then the red squares or the blue squares (or both) form a queen-connected set.

Professor Halfbrain's third theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red and blue and yellow, then the red squares or the blue squares or the yellow squares (or two of these sets, or all three sets) form a queen-connected set.

Professor Halfbrain's fourth theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red, blue, yellow and green, then at least one of the color classes must form a queen-connected set.

We all know Professor Halfbrain as an honorable and trustworthy mathematician, but even the best mathematician may sometimes be mistaken. Hence the question: Which of the Professor's four theorems do actually hold true?

Professor Halfbrain has spent his last weekend with analyzing $n\times n$ chessboards. Halfbrain says that a subset $S$ of squares on such a chessboard is queen-connected, if a chess queen can move around on $S$ and reach all the squares in $S$ while making only legal queen moves. (The queen is only allowed to move on $S$, but may visit every square in $S$ many times. Every move must start and end on a square of $S$, while the passed-through squares of a move are not necessarily in $S$.)

Professor Halfbrain's first theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red, then these red squares form a queen-connected set.

Professor Halfbrain's second theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red and blue, then the red squares or the blue squares (or both) form a queen-connected set.

Professor Halfbrain's third theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red and blue and yellow, then the red squares or the blue squares or the yellow squares (or two of these sets, or all three sets) form a queen-connected set.

Professor Halfbrain's fourth theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red, blue, yellow and green, then at least one of the color classes must form a queen-connected set.

We all know Professor Halfbrain as an honorable and trustworthy mathematician, but even the best mathematician may sometimes be mistaken. Hence the question: Which of the Professor's four theorems do actually hold true?

Professor Halfbrain has spent his last weekend with analyzing $n\times n$ chessboards. Halfbrain says that a subset $S$ of squares on such a chessboard is queen-connected, if a chess queen can move around on $S$ and reach all the squares in $S$ while making only legal queen moves. (The queen is only allowed to move from squares in $S$ to squares in $S$, but the move may pass over other squares that are not necessarily in $S$. Every square in $S$ may be visited many times.)

Professor Halfbrain's first theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red, then these red squares form a queen-connected set.

Professor Halfbrain's second theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red and blue, then the red squares or the blue squares (or both) form a queen-connected set.

Professor Halfbrain's third theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red and blue and yellow, then the red squares or the blue squares or the yellow squares (or two of these sets, or all three sets) form a queen-connected set.

Professor Halfbrain's fourth theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red, blue, yellow and green, then at least one of the color classes must form a queen-connected set.

We all know Professor Halfbrain as an honorable and trustworthy mathematician, but even the best mathematician may sometimes be mistaken. Hence the question: Which of the Professor's four theorems do actually hold true?

Professor Halfbrain has spent his last weekend with analyzing $n\times n$ chessboards. Halfbrain says that a subset $S$ of squares on such a chessboard is queen-connected, if a chess queen can move around on $S$ and reach all the squares in $S$ while making only legal queen moves. (The queen is only allowed to move on $S$, but may visit every square in $S$ many times.)

Professor Halfbrain's first theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red, then these red squares form a queen-connected set.

Professor Halfbrain's second theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red and blue, then the red squares or the blue squares (or both) form a queen-connected set.

Professor Halfbrain's third theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red and blue and yellow, then the red squares or the blue squares or the yellow squares (or two of these sets, or all three sets) form a queen-connected set.

Professor Halfbrain's fourth theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red, blue, yellow and green, then at least one of the color classes must form a queen-connected set.

We all know Professor Halfbrain as an honorable and trustworthy mathematician, but even the best mathematician may sometimes be mistaken. Hence the question: Which of the Professor's four theorems do actually hold true?

Professor Halfbrain has spent his last weekend with analyzing $n\times n$ chessboards. Halfbrain says that a subset $S$ of squares on such a chessboard is queen-connected, if a chess queen can move around on $S$ and reach all the squares in $S$ while making only legal queen moves. (The queen is only allowed to move on $S$, but may visit every square in $S$ many times. Every move must start and end on a square of $S$, while the passed-through squares of a move are not necessarily in $S$.)

Professor Halfbrain's first theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red, then these red squares form a queen-connected set.

Professor Halfbrain's second theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red and blue, then the red squares or the blue squares (or both) form a queen-connected set.

Professor Halfbrain's third theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red and blue and yellow, then the red squares or the blue squares or the yellow squares (or two of these sets, or all three sets) form a queen-connected set.

Professor Halfbrain's fourth theorem: If the squares of an $n\times n$ chessboard with $n\ge2$ are painted red, blue, yellow and green, then at least one of the color classes must form a queen-connected set.

We all know Professor Halfbrain as an honorable and trustworthy mathematician, but even the best mathematician may sometimes be mistaken. Hence the question: Which of the Professor's four theorems do actually hold true?