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Pandiagonal magic square
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==4Γ4 pandiagonal magic squares== [[File:4x4_magic_square_hierarchy.svg|thumb|upright|[[Euler diagram]] of properties of some types of 4{{times}}4 magic squares. Cells of the same colour sum to the magic constant.]] The smallest non-trivial pandiagonal magic squares are 4{{times}}4 squares. All 4{{times}}4 pandiagonal magic squares must be [[translational symmetry|translationally symmetric]] to the form<ref>{{cite web | url = https://matthbeck.github.io/teach/masters/louis.pdf | title = Magic Counting with Inside-Out Polytopes | date = May 13, 2018 | first = Louis | last = Ng}}</ref> {|class="wikitable" style="margin-left: auto; margin-right: auto; border: none; text-align:center; width:27em; height:27em; table-layout:fixed;" |- | ''a'' || ''a''+''b''+''c''+''e'' || ''a''+''c''+''d'' || ''a''+''b''+''d''+''e'' |- | ''a''+''b''+''c''+''d'' || ''a''+''d''+''e'' || ''a''+''b'' || ''a''+''c''+''e'' |- | ''a''+''b''+''e'' || ''a''+''c'' || ''a''+''b''+''c''+''d''+''e'' || ''a''+''d'' |- | ''a''+''c''+''d''+''e'' || ''a''+''b''+''d'' || ''a''+''e'' || ''a''+''b''+''c'' |} Since each 2{{times}}2 subsquare sums to the magic constant, 4{{times}}4 pandiagonal magic squares are [[most-perfect magic square]]s. In addition, the two numbers at the opposite corners of any 3{{times}}3 square add up to half the magic constant. Consequently, all 4{{times}}4 pandiagonal magic squares that are [[associative magic square|associative]] must have duplicate cells. All 4{{times}}4 pandiagonal magic squares using numbers 1-16 without duplicates are obtained by letting {{mvar|a}} equal 1; letting {{mvar|b}}, {{mvar|c}}, {{mvar|d}}, and {{mvar|e}} equal 1, 2, 4, and 8 in some order; and applying some [[translation (geometry)|translation]]. For example, with {{math|1=''b'' = 1}}, {{math|1=''c'' = 2}}, {{math|1=''d'' = 4}}, and {{math|1=''e'' = 8}}, we have the magic square {|class="wikitable" style="margin-left: auto; margin-right: auto; border: none; text-align:center; text-align:center; width:8em; height:8em; table-layout:fixed;" |- | 1 || 12 || 7 || 14 |- | 8 || 13 || 2 || 11 |- | 10 || 3 || 16 || 5 |- | 15 || 6 || 9 || 4 |} The number of 4{{times}}4 pandiagonal magic squares using numbers 1-16 without duplicates is 384 (16 times 24, where 16 accounts for the translation and 24 accounts for the 4[[factorial|!]] ways to assign 1, 2, 4, and 8 to {{mvar|b}}, {{mvar|c}}, {{mvar|d}}, and {{mvar|e}}).
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