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KingGeorge
 2 years ago
[UNSOLVED] Combinatorics challenge problem!
Suppose you have a chessboard of size \(n\times n\), and you want to place some pawns on it. How many ways are there to place these pawns such that each row and column has an even number of pawns?
KingGeorge
 2 years ago
[UNSOLVED] Combinatorics challenge problem! Suppose you have a chessboard of size \(n\times n\), and you want to place some pawns on it. How many ways are there to place these pawns such that each row and column has an even number of pawns?

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vf321
 2 years ago
Best ResponseYou've already chosen the best response.0Does chessboard orientation matter?

vf321
 2 years ago
Best ResponseYou've already chosen the best response.0OK, well I don't want to make you think that I'm still working on it. I tried for a couple days, but then I gave up.

andriod09
 2 years ago
Best ResponseYou've already chosen the best response.0wouldn't you have to fill the entire board so that way every column and every row have the same number of pawns in it?

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0Note necessarily. For example, in a 4x4 chessboard, you can have a 4x2 rectangle filled with pawns.

Numb3r1
 2 years ago
Best ResponseYou've already chosen the best response.0I'm going to try a few n's and see what patterns I notice. n=1, x=0 (1 if the zeroboard counts) n=2, x=1 (2 if the zeroboard counts) n=3, x=3 (4 if the zeroboard counts) assuming that orientation does not matter, therefore rotations and reflections count as the same thing. n=4, x=7 (8 if the zeroboard counts) Based on this, x=\[2^{n1}1, 2^{n1} if the zeroboard counts.\]

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0@Numb3r1 the zero board is does count, and \(2^{n1}\) is correct. Can you prove it?

Numb3r1
 2 years ago
Best ResponseYou've already chosen the best response.0I have to prove that adding one more row and one more column doubles the number of possibilities, or find a recursive relation between previous boards. The recursion option seems simpler to me. One moment...

Numb3r1
 2 years ago
Best ResponseYou've already chosen the best response.0A zerobyzero board does not exist, so it will obviously have 0 options. The onebyone board has one option. The twobytwo board has two options (and evennumbered boards will always have a new option in terms of how many pieces are in a row). Adding one row and one column means adding one new option for each option that was available on the previous board (one expanding sideways, one expanding both sideways and upwards). From the 3x3 board, the 4x4 has the same doubleexpansion factor on the square subboards (the 2x2 board and 3x3 with only the corners) but the two rectangular subboards can only be lengthened (expanding upwards will lead to the same result as another rectangle or square being lengthened). The 2x2 board in the middle and the 4x4 board are the new options corresponding to the rectangles... This approach isn't leading to an elegant proof, so I'll have to reevaluate.

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0I think your method could lead to a relatively good proof, but it's definitely not what I ended up with. Let me know if you want a hint about my method.

Numb3r1
 2 years ago
Best ResponseYou've already chosen the best response.0\[\sum_{1}^{n}n!+(n1)!+....+1!\] seems to work.

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0I'm not sure I understand what that's supposed to be doing.

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0@Numb3r1 I apologize, I misread my previous work. \(2^{n1}\) is not the correct answer. It should be \[\large 2^{(n1)^2}.\] This is because we consider a rotation of the board to potentially be a different solution.

Numb3r1
 2 years ago
Best ResponseYou've already chosen the best response.0I had already ignored rotations, and that means that I was correct as well! Actually, are you sure it wasn't \[2^{2(n1)}\]? I don't see how a 7x7 board would have six times as many options if you can rotate it.

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0I am absolutely positive this is the correct solution. Consider the smaller \((n1)\times(n1)\) chessboard that is a subboard of the larger one, and start putting pawns on that. What can you say about the placement of pawns on the rest of the board?

Numb3r1
 2 years ago
Best ResponseYou've already chosen the best response.0My point is that it could be \[(2^{n1})^{2}=2^{2(n1)} but 2^{(n1)^{2}}wont work.\]

Numb3r1
 2 years ago
Best ResponseYou've already chosen the best response.0To clarify, the first one will, the second will not.

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0A 7x7 board will have \(2^{36}\) possible solutions, not \(2^{12}\) or \(2^6\). Again, I apologize for leading you down the wrong path for a while :(

Numb3r1
 2 years ago
Best ResponseYou've already chosen the best response.0For a 3x3 board, \[\left[\begin{matrix} 1& 0& 1 \\ 0&0 &0\\ 1 &0 &1 \end{matrix}\right],\left[\begin{matrix} 1& 1& 0 \\ 1& 1&0\\ 0 &0 & 0\end{matrix}\right]\times4,\left[\begin{matrix} 1&0 & 1\\ 1& 0&1\\ 0&0 &0 \end{matrix}\right]\times4,\left[\begin{matrix} 0& 0&0 \\ 0& 0&0\\ 0 &0 & 0\end{matrix}\right], and what else?\]

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0\[\begin{bmatrix}0&1&1\\1&1&0\\ 1&0&1\end{bmatrix}\]for example.

Numb3r1
 2 years ago
Best ResponseYou've already chosen the best response.0Ah, that times four, and then there are two more? I'll have to try another approach to the problem.

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0the last two I believe are given by \[\begin{bmatrix}1&1&0\\1&0&1\\0&1&1\end{bmatrix}\]and a 90 degree rotation.

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0Well, I've got to sleep now, but I would strongly consider looking at the smaller board of dimensions \((n1)\times(n1)\) (which has \((n1)^2\) boxes), placing some pawns on that, and then looking at where you can place pawns on the one row/column not included in this smaller board.

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.0Wait for 5 minutes more kg, I am trying.

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0I should correct what I said near the top. Orientation does actually matter. I just had a slight misunderstanding when I read vf321's comment.

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.0What should be the correct answer @KingGeorge ?

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0\[\large 2^{(n1)^2}\](The \(n1\) is squared)

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.0But if n = 1, then it is 1, but it can not exist ?

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0If \(n=1\), then \(2^{(n1)^2}=2^0=1\). The only way would be to have the empty board.

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.0lol, empty board also allowed :
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