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Pandiagonal magic square
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== (4''n''+2)Γ(4''n''+2) pandiagonal magic squares with nonconsecutive elements == No pandiagonal magic square exists of order <math>4n+2</math> if consecutive [[integer]]s are used. But certain sequences of nonconsecutive integers do admit order-(<math>4n+2</math>) pandiagonal magic squares. Consider the sum 1+2+3+5+6+7 = 24. This sum can be divided in half by taking the appropriate groups of three addends, or in thirds using groups of two addends: : 1+5+6 = 2+3+7 = 12 : 1+7 = 2+6 = 3+5 = 8 An additional equal partitioning of the sum of squares guarantees the semi-bimagic property noted below: : 1<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup> = 2<sup>2</sup> + 3<sup>2</sup> + 7<sup>2</sup> = 62 Note that the consecutive integer sum 1+2+3+4+5+6 = 21, an [[parity (mathematics)|odd]] sum, lacks the half-partitioning. With both equal partitions available, the numbers 1, 2, 3, 5, 6, 7 can be arranged into 6 Γ 6 pandigonal patterns {{mvar|A}} and {{mvar|B}}, respectively given by: {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:12em;height:12em;table-layout:fixed;" |- | 1 || 5 || 6 || 7 || 3 || 2 |- | 5 || 6 || 1 || 3 || 2 || 7 |- | 6 || 1 || 5 || 2 || 7 || 3 |- | 1 || 5 || 6 || 7 || 3 || 2 |- | 5 || 6 || 1 || 3 || 2 || 7 |- | 6 || 1 || 5 || 2 || 7 || 3 |} {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:12em;height:12em;table-layout:fixed;" | 6 || 5 || 1 || 6 || 5 || 1 |- | 1 || 6 || 5 || 1 || 6 || 5 |- | 5 || 1 || 6 || 5 || 1 || 6 |- | 2 || 3 || 7 || 2 || 3 || 7 |- | 7 || 2 || 3 || 7 || 2 || 3 |- | 3 || 7 || 2 || 3 || 7 || 2 |} Then <math>7A + B - 7C</math> (where {{mvar|C}} is the magic square with 1 for all cells) gives the nonconsecutive pandiagonal 6 Γ 6 square: {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:12em;height:12em;table-layout:fixed;" |- | 6 || 33 || 36 || 48 || 19 || 8 |- | 29 || 41 || 5 || 15 || 13 || 47 |- | 40 || 1 || 34 || 12 || 43 || 20 |- | 2 || 31 || 42 || 44 || 17 || 14 |- | 35 || 37 || 3 || 21 || 9 || 45 |- | 38 || 7 || 30 || 10 || 49 || 16 |} with a maximum element of 49 and a pandiagonal magic constant of 150. This square is pandiagonal and semi-bimagic, that means that rows, columns, main diagonals and broken diagonals have a sum of 150 and, if we square all the numbers in the square, only the rows and the columns are magic and have a sum of 5150. For 10th order a similar construction is possible using the equal partitionings of the sum 1+2+3+4+5+9+10+11+12+13 = 70: : 1+3+9+10+12 = 2+4+5+11+13 = 35 : 1+13 = 2+12 = 3+11 = 4+10 = 5+9 = 14 : 1<sup>2</sup> + 3<sup>2</sup> + 9<sup>2</sup> + 10<sup>2</sup> + 12<sup>2</sup> = 2<sup>2</sup> + 4<sup>2</sup> + 5<sup>2</sup> + 11<sup>2</sup> + 13<sup>2</sup> = 335 (equal partitioning of squares; semi-bimagic property) This leads to squares having a maximum element of 169 and a pandiagonal magic constant of 850, which are also semi-bimagic with each row or column sum of squares equal to 102,850.
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