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Pandiagonal magic square
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==5Γ5 pandiagonal magic squares== There are many 5 Γ 5 pandiagonal magic squares. Unlike 4 Γ 4 pandiagonal magic squares, these can be [[associative magic square|associative]]. The following is a 5 Γ 5 associative pandiagonal magic square: {|class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:10em;height:10em;table-layout:fixed;" |- | 20 || 8 || 21 || 14 || 2 |- | 11 || 4 || 17 || 10 || 23 |- | 7 || 25 || 13 || 1 || 19 |- | 3 || 16 || 9 || 22 || 15 |- | 24 || 12 || 5 || 18 || 6 |} In addition to the rows, columns, and diagonals, a 5 Γ 5 pandiagonal magic square also shows its magic constant in four "[[quincunx]]" patterns, which in the above example are: : 17+25+13+1+9 = 65 (center plus adjacent row and column squares) : 21+7+13+19+5 = 65 (center plus the remaining row and column squares) : 4+10+13+16+22 = 65 (center plus diagonally adjacent squares) : 20+2+13+24+6 = 65 (center plus the remaining squares on its diagonals) Each of these quincunxes can be translated to other positions in the square by [[cyclic permutation]] of the rows and columns (wrapping around), which in a pandiagonal magic square does not affect the equality of the magic constants. This leads to 100 quincunx sums, including broken quincunxes analogous to broken diagonals. The quincunx sums can be proved by taking [[linear combination]]s of the row, column, and diagonal sums. Consider the pandiagonal magic square :<math>\begin{array}{|c|c|c|c|c|} \hline \!\!\!\; a_{11} \!\!\! & \!\! a_{12} \!\!\! & \!\! a_{13} \!\!\! & \!\! a_{14} \!\!\! & \!\! a_{15} \!\!\\ \hline \!\!\!\; a_{21} \!\!\! & \!\! a_{22} \!\!\! & \!\! a_{23} \!\!\! & \!\! a_{24} \!\!\! & \!\! a_{25} \!\!\\ \hline \!\!\!\; a_{31} \!\!\! & \!\! a_{32} \!\!\! & \!\! a_{33} \!\!\! & \!\! a_{34} \!\!\! & \!\! a_{35} \!\!\\ \hline \!\!\!\; a_{41} \!\!\! & \!\! a_{42} \!\!\! & \!\! a_{43} \!\!\! & \!\! a_{44} \!\!\! & \!\! a_{45} \!\!\\ \hline \!\!\!\; a_{51} \!\!\! & \!\! a_{52} \!\!\! & \!\! a_{53} \!\!\! & \!\! a_{54} \!\!\! & \!\! a_{55} \!\!\\ \hline \end{array}</math> with magic constant {{mvar|s}}. To prove the quincunx sum <math>a_{11} + a_{15} + a_{33} + a_{51} + a_{55} = s</math> (corresponding to the 20+2+13+24+6 = 65 example given above), we can add together the following: : 3 times each of the diagonal sums <math>a_{11} + a_{22} + a_{33} + a_{44} + a_{55}</math> and <math>a_{15} + a_{24} + a_{33} + a_{42} + a_{51}</math>, : The diagonal sums <math>a_{11} + a_{25} + a_{34} + a_{43} + a_{52}</math>, <math>a_{12} + a_{23} + a_{34} + a_{45} + a_{51}</math>, <math>a_{14} + a_{23} + a_{32} + a_{41} + a_{55}</math>, and <math>a_{15} + a_{21} + a_{32} + a_{43} + a_{54}</math>, : The row sums <math>a_{11} + a_{12} + a_{13} + a_{14} + a_{15}</math> and <math>a_{51} + a_{52} + a_{53} + a_{54} + a_{55}</math>. From this sum, subtract the following: : The row sums <math>a_{21} + a_{22} + a_{23} + a_{24} + a_{25}</math> and <math>a_{41} + a_{42} + a_{43} + a_{44} + a_{45}</math>, : The column sum <math>a_{13} + a_{23} + a_{33} + a_{43} + a_{53}</math>, : Twice each of the column sums <math>a_{12} + a_{22} + a_{32} + a_{42} + a_{52}</math> and <math>a_{14} + a_{24} + a_{34} + a_{44} + a_{54}</math>. The net result is <math>5a_{11} + 5a_{15} + 5a_{33} + 5a_{51} + 5a_{55} = 5s</math>, which divided by 5 gives the quincunx sum. Similar linear combinations can be constructed for the other quincunx patterns <math>a_{23} + a_{32} + a_{33} + a_{34} + a_{43}</math>, <math>a_{13} + a_{31} + a_{33} + a_{35} + a_{53}</math>, and <math>a_{22} + a_{24} + a_{33} + a_{42} + a_{44}</math>.
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