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How many ways can a domino be placed on a 4x4 chessboard? Each "half" of the domino must cover exactly one square of the chessboard. 112 is not answer
 

Guest May 1, 2018

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 #1
avatar+19344 
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How many ways can a domino be placed on a 4x4 chessboard?

Each "half" of the domino must cover exactly one square of the chessboard.

 

There is a general formula:

The Number of domino tilings (or dimer coverings) of a 2n X 2n square is:

\(\displaystyle \prod \limits_{j=1}^{n}\prod \limits_{k=1}^{n} \left[ 4*\cos^2\left( \frac{j* \pi}{2*n+1} \right) + 4*\cos^2\left( \frac{k* \pi}{2*n+1} \right) \right]\)

 

Covering a 4 x 4 Chessboard with Dominoes: \(n = 2\)

\(\begin{array}{|rcll|} \hline \displaystyle \prod \limits_{j=1}^{2}\prod \limits_{k=1}^{2} \left[ 4*\cos^2\left( \frac{j* \pi}{5} \right) + 4*\cos^2\left( \frac{k* \pi}{5} \right) \right] = 36 \\ \hline \end{array} \)

 

Ther are 36 ways a domino can be placed on a 4x4 chessboard.

 

The 36 solutions for the 4 X 4 board:

\(\begin{array}{|rcll|} \hline A01 = {(1,2), (3,4), (5,6), (7,8), (9,10), (11,12), (13,14), (15,16)} \\ A02 = {(1,2), (3,4), (5,6), (7,11), (9,10), (8,12), (13,14), (15,16)}\\ A03 = {(1,2), (3,4), (5,9), (6,7), (10,11), (8,12), (13,14), (15,16)}\\ A04 = {(1,2), (3,4), (5,9), (6,10), (7,8), (11,12), (13,14), (15,16)}\\ A05 = {(1,2), (3,4), (5,9), (6,10), (7,11), (8,12), (13,14), (15,16)}\\ A06 = {(1,2), (3,4), (5,6), (7,8), (9,10), (13,14), (11,15), (12,16)}\\ A07 = {(1,2), (3,4), (5,9), (6,10), (7,8), (11,15), (13,14), (12,16)}\\ A08 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,14), (11,12), (15,16)}\\ A09 = {(1,2), (3,4), (5,6), (7,11), (8,12), (9,13), (10,14), (15,16)}\\ A10 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,11), (14,15), (12,16)}\\ A11 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,14), (11,15), (12,16)}\\ A12 = {(1,2), (5,6), (3,7), (4,8), (9,10), (11,12), (13,14), (15,16)}\\ A13 = {(1,2), (3,7), (4,8), (5,9), (6,10), (11,12), (13,14), (15,16)}\\ A14 = {(1,2), (5,6), (3,7), (4,8), (9,10), (13,14), (11,15), (12,16)}\\ A15 = {(1,2), (3,7), (4,8), (6,10), (5,9), (11,15), (12,16), (13,14)}\\ A16 = {(1,2), (3,7), (4,8), (5,6), (9,13), (10,14), (11,12), (15,16)}\\ A17 = {(1,2), (3,7), (4,8), (5,6), (9,13), (10,11), (14,15), (12,16)}\\ A18 = {(1,2), (5,6), (3,7), (4,8), (9,13), (10,14), (11,15), (12,16)}\\ A19 = {(1,5), (2,6), (3,4), (7,8), (9,10), (11,12), (13,14), (15,16)}\\ A20 = {(1,5), (2,6), (3,4), (7,11), (8,12), (9,10), (13,14), (15,16)}\\ A21 = {(1,5), (3,4), (2,6), (9,10), (7,8), (11,15), (13,14), (12,16)}\\ A22 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,14), (11,12), (15,16)}\\ A23 = {(1,5), (2,6), (3,4), (7,11), (8,12), (9,13), (10,14), (15,16)}\\ A24 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,11), (14,15), (12,16)}\\ A25 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,14), (11,15), (12,16)}\\ A26 = {(1,5), (2,3), (6,7), (4,8), (9,10), (11,12), (13,14), (15,16)}\\ A27 = {(1,5), (2,6), (3,7), (4,8), (9,10), (11,12), (13,14), (15,16)}\\ A28 = {(1,5), (2,3), (6,7), (4,8), (9,10), (11,15), (13,14), (12,16)}\\ A29 = {(1,5), (2,6), (3,7), (4,8), (9,10), (13,14), (11,15), (12,16)}\\ A30 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,14), (11,12), (15,16)}\\ A31 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,14), (11,12), (15,16)}\\ A32 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,14), (11,15), (12,16)}\\ A33 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,11), (14,15), (12,16)}\\ A34 = {(1,5), (2,3), (4,8), (6,10), (7,11), (9,13), (14,15), (12,16)}\\ A35 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,14), (11,15), (12,16)}\\ A36 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,11), (14,15), (12,16)} \\ \hline \end{array}\)

 

with \(\begin{array}{|r|r|r|r|} \hline 13 & 14 & 15 & 16 \\ \hline 9 & 10 & 11 & 12 \\ \hline 5 & 6 & 7 & 8 \\ \hline 1 & 2 & 3 & 4 \\ \hline \end{array}\)

 

 

laugh

heureka  May 2, 2018
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1+0 Answers

 #1
avatar+19344 
+1
Best Answer

How many ways can a domino be placed on a 4x4 chessboard?

Each "half" of the domino must cover exactly one square of the chessboard.

 

There is a general formula:

The Number of domino tilings (or dimer coverings) of a 2n X 2n square is:

\(\displaystyle \prod \limits_{j=1}^{n}\prod \limits_{k=1}^{n} \left[ 4*\cos^2\left( \frac{j* \pi}{2*n+1} \right) + 4*\cos^2\left( \frac{k* \pi}{2*n+1} \right) \right]\)

 

Covering a 4 x 4 Chessboard with Dominoes: \(n = 2\)

\(\begin{array}{|rcll|} \hline \displaystyle \prod \limits_{j=1}^{2}\prod \limits_{k=1}^{2} \left[ 4*\cos^2\left( \frac{j* \pi}{5} \right) + 4*\cos^2\left( \frac{k* \pi}{5} \right) \right] = 36 \\ \hline \end{array} \)

 

Ther are 36 ways a domino can be placed on a 4x4 chessboard.

 

The 36 solutions for the 4 X 4 board:

\(\begin{array}{|rcll|} \hline A01 = {(1,2), (3,4), (5,6), (7,8), (9,10), (11,12), (13,14), (15,16)} \\ A02 = {(1,2), (3,4), (5,6), (7,11), (9,10), (8,12), (13,14), (15,16)}\\ A03 = {(1,2), (3,4), (5,9), (6,7), (10,11), (8,12), (13,14), (15,16)}\\ A04 = {(1,2), (3,4), (5,9), (6,10), (7,8), (11,12), (13,14), (15,16)}\\ A05 = {(1,2), (3,4), (5,9), (6,10), (7,11), (8,12), (13,14), (15,16)}\\ A06 = {(1,2), (3,4), (5,6), (7,8), (9,10), (13,14), (11,15), (12,16)}\\ A07 = {(1,2), (3,4), (5,9), (6,10), (7,8), (11,15), (13,14), (12,16)}\\ A08 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,14), (11,12), (15,16)}\\ A09 = {(1,2), (3,4), (5,6), (7,11), (8,12), (9,13), (10,14), (15,16)}\\ A10 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,11), (14,15), (12,16)}\\ A11 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,14), (11,15), (12,16)}\\ A12 = {(1,2), (5,6), (3,7), (4,8), (9,10), (11,12), (13,14), (15,16)}\\ A13 = {(1,2), (3,7), (4,8), (5,9), (6,10), (11,12), (13,14), (15,16)}\\ A14 = {(1,2), (5,6), (3,7), (4,8), (9,10), (13,14), (11,15), (12,16)}\\ A15 = {(1,2), (3,7), (4,8), (6,10), (5,9), (11,15), (12,16), (13,14)}\\ A16 = {(1,2), (3,7), (4,8), (5,6), (9,13), (10,14), (11,12), (15,16)}\\ A17 = {(1,2), (3,7), (4,8), (5,6), (9,13), (10,11), (14,15), (12,16)}\\ A18 = {(1,2), (5,6), (3,7), (4,8), (9,13), (10,14), (11,15), (12,16)}\\ A19 = {(1,5), (2,6), (3,4), (7,8), (9,10), (11,12), (13,14), (15,16)}\\ A20 = {(1,5), (2,6), (3,4), (7,11), (8,12), (9,10), (13,14), (15,16)}\\ A21 = {(1,5), (3,4), (2,6), (9,10), (7,8), (11,15), (13,14), (12,16)}\\ A22 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,14), (11,12), (15,16)}\\ A23 = {(1,5), (2,6), (3,4), (7,11), (8,12), (9,13), (10,14), (15,16)}\\ A24 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,11), (14,15), (12,16)}\\ A25 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,14), (11,15), (12,16)}\\ A26 = {(1,5), (2,3), (6,7), (4,8), (9,10), (11,12), (13,14), (15,16)}\\ A27 = {(1,5), (2,6), (3,7), (4,8), (9,10), (11,12), (13,14), (15,16)}\\ A28 = {(1,5), (2,3), (6,7), (4,8), (9,10), (11,15), (13,14), (12,16)}\\ A29 = {(1,5), (2,6), (3,7), (4,8), (9,10), (13,14), (11,15), (12,16)}\\ A30 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,14), (11,12), (15,16)}\\ A31 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,14), (11,12), (15,16)}\\ A32 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,14), (11,15), (12,16)}\\ A33 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,11), (14,15), (12,16)}\\ A34 = {(1,5), (2,3), (4,8), (6,10), (7,11), (9,13), (14,15), (12,16)}\\ A35 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,14), (11,15), (12,16)}\\ A36 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,11), (14,15), (12,16)} \\ \hline \end{array}\)

 

with \(\begin{array}{|r|r|r|r|} \hline 13 & 14 & 15 & 16 \\ \hline 9 & 10 & 11 & 12 \\ \hline 5 & 6 & 7 & 8 \\ \hline 1 & 2 & 3 & 4 \\ \hline \end{array}\)

 

 

laugh

heureka  May 2, 2018

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