Editing Strachey method for magic squares

{{One source|date=September 2024}}
The '''Strachey method for magic squares''' is an [[algorithm]] for generating [[magic square]]s of [[singly even]] order 4''k'' + 2. An example of magic square of order 6 constructed with the Strachey method:

{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:20em;height:20em;table-layout:fixed;"
|-
! colspan="6" | Example
|-
|35 || 1 || 6 || 26 || 19 || 24
|-
| 3 || 32 || 7 || 21 || 23 || 25
|-
| 31 || 9 || 2 || 22 || 27 || 20
|-
| 8 || 28 || 33 || 17 || 10 || 15
|-
| 30 || 5 || 34 || 12 || 14 || 16
|-
| 4 || 36 || 29 || 13 || 18 || 11
|}

Strachey's method of construction of singly even magic square of order ''n'' = 4''k'' + 2.

'''1.''' Divide the grid into 4 quarters each having ''n''<sup>2</sup>/4 cells and name them crosswise thus

{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:4em;height:4em;table-layout:fixed;"
|-
|A || C
|-
|D || B
|}

'''2.''' Using the [[Siamese method]] (De la Loubère method) complete the individual magic squares of odd order 2''k'' + 1 in subsquares '''A''', '''B''', '''C''', '''D''', first filling up the sub-square '''A''' with the numbers 1 to ''n''<sup>2</sup>/4, then the sub-square '''B''' with the numbers ''n''<sup>2</sup>/4 + 1 to 2''n''<sup>2</sup>/4,then the sub-square '''C''' with the numbers 2''n''<sup>2</sup>/4 + 1 to 3''n''<sup>2</sup>/4,  then the sub-square '''D''' with the numbers 3''n''<sup>2</sup>/4 + 1 to ''n''<sup>2</sup>. As a running example, we consider a 10×10 magic square, where we have divided the square into four quarters. The quarter '''A''' contains a magic square of numbers from 1 to 25, '''B''' a magic square of numbers from 26 to 50, '''C''' a magic square of numbers from 51 to 75, and '''D''' a magic square of numbers from 76 to 100.  

{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:30em;height:30em;table-layout:fixed;"
|-
| style="background-color: silver;"|17 || style="background-color: silver;"|24 ||   style="background-color: silver;"|1 || style="background-color: silver;"|8 || style="background-color: silver;"|15 || 67 || 74 || 51 || 58 || 65
|-
| style="background-color: silver;"|23 ||  style="background-color: silver;"|5 ||   style="background-color: silver;"|7 || style="background-color: silver;"|14 || style="background-color: silver;"|16 || 73 || 55 || 57 || 64 || 66
|-
| style="background-color: silver;"|4 ||  style="background-color: silver;"|6 ||  style="background-color: silver;"|13 || style="background-color: silver;"|20 || style="background-color: silver;"|22 || 54 || 56 || 63 || 70 || 72
|-
| style="background-color: silver;"|10 || style="background-color: silver;"|12 ||  style="background-color: silver;"|19 || style="background-color: silver;"|21 ||  style="background-color: silver;"|3 || 60 || 62 || 69 || 71 || 53 
|-
| style="background-color: silver;"|11 || style="background-color: silver;"|18 ||  style="background-color: silver;"|25 ||  style="background-color: silver;"|2 ||  style="background-color: silver;"|9 || 61 || 68 || 75 || 52 || 59 
|-
| 92 || 99 ||  76 || 83 || 90 || style="background-color: silver;"|42 || style="background-color: silver;"|49 || style="background-color: silver;"|26 || style="background-color: silver;"|33 || style="background-color: silver;"|40
|-
| 98 || 80 ||  82 || 89 || 91 || style="background-color: silver;"|48 || style="background-color: silver;"|30 || style="background-color: silver;"|32 || style="background-color: silver;"|39 || style="background-color: silver;"|41
|-
| 79 || 81 ||  88 || 95 || 97 || style="background-color: silver;"|29 || style="background-color: silver;"|31 || style="background-color: silver;"|38 || style="background-color: silver;"|45 || style="background-color: silver;"|47
|-
| 85 || 87 ||  94 || 96 || 78 || style="background-color: silver;"|35 || style="background-color: silver;"|37 || style="background-color: silver;"|44 || style="background-color: silver;"|46 || style="background-color: silver;"|28
|-
| 86 || 93 || 100 || 77 || 84 || style="background-color: silver;"|36 || style="background-color: silver;"|43 || style="background-color: silver;"|50 || style="background-color: silver;"|27 || style="background-color: silver;"|34
|}

'''3.''' Exchange the leftmost '''k''' columns in sub-square '''A''' with the corresponding columns of sub-square '''D'''.
				
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:30em;height:30em;table-layout:fixed;"
|-											
| '''92''' || '''''99''''' || 1 || 8 || 15 || 67 || 74 || 51 || 58 || 65
|-
|'''98''' || '''80''' || 7 || 14 || 16 || 73 || 55 || 57 || 64 || 66
|-
|'''79''' || '''81''' || 13 || 20 || 22 || 54 || 56 || 63 || 70 || 72
|-
|'''85''' || '''87''' || 19 || 21 || 3 || 60 || 62 || 69 || 71 || 53
|-
|'''86''' || '''93''' || 25 || 2 || 9 || 61 || 68 || 75 || 52 || 59
|-
|'''17''' || '''24''' || 76 || 83 || 90 || 42 || 49 || 26 || 33 || 40
|-
|'''23''' || '''5''' || 82 || 89 || 91 || 48 || 30 || 32 || 39 || 41
|-
|'''4''' || '''6''' || 88 || 95 || 97 || 29 || 31 || 38 || 45 || 47
|-
|'''10''' || '''12''' || 94 || 96 || 78 || 35 || 37 || 44 || 46 || 28
|-
|'''11''' || '''18''' || 100 || 77 || 84 || 36 || 43 || 50 || 27 || 34
|}
											
'''4.''' Exchange the rightmost '''k - 1''' columns in sub-square '''C''' with the corresponding columns of sub-square '''B'''.

{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:30em;height:30em;table-layout:fixed;"
|-
| 92 || 99 || 1 || 8 || 15 || 67 || 74 || 51 || 58 || '''40'''
|-
|98 || 80 || 7 || 14 || 16 || 73 || 55 || 57 || 64 || '''41'''
|-
|79 || 81 || 13 || 20 || 22 || 54 || 56 || 63 || 70 || '''47'''
|-
|85 || 87 || 19 || 21 || 3 || 60 || 62 || 69 || 71 || '''28'''
|-
|86 || 93 || 25 || 2 || 9 || 61 || 68 || 75 || 52 || '''34'''
|-
|17 || 24 || 76 || 83 || 90 || 42 || 49 || 26 || 33 || '''65'''
|-
|23 || 5 || 82 || 89 || 91 || 48 || 30 || 32 || 39 || '''66'''
|-
|4 || 6 || 88 || 95 || 97 || 29 || 31 || 38 || 45 || '''72'''
|-
|10 || 12 || 94 || 96 || 78 || 35 || 37 || 44 || 46 || '''53'''
|-
|11 || 18 || 100 || 77 || 84 || 36 || 43 || 50 || 27 || '''59'''
|}
											
'''5.''' Exchange the middle cell of the leftmost column of sub-square '''A''' with the corresponding cell of sub-square '''D'''. Exchange the central cell in sub-square '''A''' with the corresponding cell of sub-square '''D'''.
												
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:30em;height:30em;table-layout:fixed;"
|-	
|92 || 99 || 1 || 8 || 15 || 67 || 74 || 51 || 58 || 40
|-
|98 || 80 || 7 || 14 || 16 || 73 || 55 || 57 || 64 || 41
|-
|'''4''' || 81 || '''88''' || 20 || 22 || 54 || 56 || 63 || 70 || 47
|-
|85 || 87 || 19 || 21 || 3 || 60 || 62 || 69 || 71 || 28
|-
|86 || 93 || 25 || 2 || 9 || 61 || 68 || 75 || 52 || 34
|-
|17 || 24 || 76 || 83 || 90 || 42 || 49 || 26 || 33 || 65
|-
|23 || 5 || 82 || 89 || 91 || 48 || 30 || 32 || 39 || 66
|-
|'''79''' || 6 || '''13''' || 95 || 97 || 29 || 31 || 38 || 45 || 72
|-
|10 || 12 || 94 || 96 || 78 || 35 || 37 || 44 || 46 || 53
|-
|11 || 18 || 100 || 77 || 84 || 36 || 43 || 50 || 27 || 59	
|}
											
The result is a magic square of order ''n''=4''k'' + 2.<ref>W W Rouse Ball Mathematical Recreations and Essays, (1911)</ref>

==References==
{{reflist}}

==See also==
*[[Conway's LUX method for magic squares]]
*[[Siamese method]]

{{DEFAULTSORT:Strachey Method For Magic Squares}}
[[Category:Magic squares]]