Editing Pandiagonal magic square (section)

==4''n''×4''n'' pandiagonal magic squares==

A <math>4n \times 4n</math> pandiagonal magic square can be built by the following algorithm.
{{ordered list

|Put the first <math>2n</math> natural numbers into the first row and the first <math>2n</math> columns of the square. 
{{aligned table|cols=8|class=wikitable
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;
| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;
| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;
| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;
| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;
| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;
| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;
| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;
}}
|Put the next <math>2n</math> natural numbers beneath the first <math>2n</math> natural numbers in reverse. Each vertical pair must have the same sum.

{{aligned table|cols=8|class=wikitable
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;
| &nbsp; 8&nbsp; | &nbsp; 7&nbsp; | &nbsp; 6&nbsp; | &nbsp; 5&nbsp; | &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;
| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;
| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;
| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;
| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;
| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;
| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;| &nbsp;
}}
|Copy that <math>2 \times 2n</math> rectangle <math>2n-1</math> times beneath the first rectangle.

{{aligned table|cols=8|class=wikitable
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;
| &nbsp; 8&nbsp; | &nbsp; 7&nbsp; | &nbsp; 6&nbsp; | &nbsp; 5&nbsp; | &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;
| &nbsp; 8&nbsp; | &nbsp; 7&nbsp; | &nbsp; 6&nbsp; | &nbsp; 5&nbsp; | &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;
| &nbsp; 8&nbsp; | &nbsp; 7&nbsp; | &nbsp; 6&nbsp; | &nbsp; 5&nbsp; | &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;
| &nbsp; 8&nbsp; | &nbsp; 7&nbsp; | &nbsp; 6&nbsp; | &nbsp; 5&nbsp; | &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;| &nbsp; &nbsp; &nbsp;
}}
|Copy the left <math>4n \times 2n</math> rectangle into the right <math>4n \times 2n</math> rectangle but shift it ring-wise by one row.

{{aligned table|cols=8|class=wikitable
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; 8&nbsp;| &nbsp; 7&nbsp;| &nbsp; 6&nbsp;| &nbsp; 5&nbsp;
| &nbsp; 8&nbsp; | &nbsp; 7&nbsp; | &nbsp; 6&nbsp; | &nbsp; 5&nbsp; | &nbsp; 1&nbsp;| &nbsp; 2&nbsp;| &nbsp; 3&nbsp;| &nbsp; 4&nbsp;
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; 8&nbsp;| &nbsp; 7&nbsp;| &nbsp; 6&nbsp;| &nbsp; 5&nbsp;
| &nbsp; 8&nbsp; | &nbsp; 7&nbsp; | &nbsp; 6&nbsp; | &nbsp; 5&nbsp; | &nbsp; 1&nbsp;| &nbsp; 2&nbsp;| &nbsp; 3&nbsp;| &nbsp; 4&nbsp;
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; 8&nbsp;| &nbsp; 7&nbsp;| &nbsp; 6&nbsp;| &nbsp; 5&nbsp;
| &nbsp; 8&nbsp; | &nbsp; 7&nbsp; | &nbsp; 6&nbsp; | &nbsp; 5&nbsp; | &nbsp; 1&nbsp;| &nbsp; 2&nbsp;| &nbsp; 3&nbsp;| &nbsp; 4&nbsp;
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; 8&nbsp;| &nbsp; 7&nbsp;| &nbsp; 6&nbsp;| &nbsp; 5&nbsp;
| &nbsp; 8&nbsp; | &nbsp; 7&nbsp; | &nbsp; 6&nbsp; | &nbsp; 5&nbsp; | &nbsp; 1&nbsp;| &nbsp; 2&nbsp;| &nbsp; 3&nbsp;| &nbsp; 4&nbsp;
}}
| Build a second <math>4n \times 4n</math> square and copy the first square into it but turn it by 90°.
{{aligned table|cols=2|class=wikitable
| {{mvar|A}}
{{aligned table|cols=8|class=wikitable
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; 8&nbsp;| &nbsp; 7&nbsp;| &nbsp; 6&nbsp;| &nbsp; 5&nbsp;
| &nbsp; 8&nbsp; | &nbsp; 7&nbsp; | &nbsp; 6&nbsp; | &nbsp; 5&nbsp; | &nbsp; 1&nbsp;| &nbsp; 2&nbsp;| &nbsp; 3&nbsp;| &nbsp; 4&nbsp;
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; 8&nbsp;| &nbsp; 7&nbsp;| &nbsp; 6&nbsp;| &nbsp; 5&nbsp;
| &nbsp; 8&nbsp; | &nbsp; 7&nbsp; | &nbsp; 6&nbsp; | &nbsp; 5&nbsp; | &nbsp; 1&nbsp;| &nbsp; 2&nbsp;| &nbsp; 3&nbsp;| &nbsp; 4&nbsp;
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; 8&nbsp;| &nbsp; 7&nbsp;| &nbsp; 6&nbsp;| &nbsp; 5&nbsp;
| &nbsp; 8&nbsp; | &nbsp; 7&nbsp; | &nbsp; 6&nbsp; | &nbsp; 5&nbsp; | &nbsp; 1&nbsp;| &nbsp; 2&nbsp;| &nbsp; 3&nbsp;| &nbsp; 4&nbsp;
| &nbsp; 1&nbsp; | &nbsp; 2&nbsp; | &nbsp; 3&nbsp; | &nbsp; 4&nbsp; | &nbsp; 8&nbsp;| &nbsp; 7&nbsp;| &nbsp; 6&nbsp;| &nbsp; 5&nbsp;
| &nbsp; 8&nbsp; | &nbsp; 7&nbsp; | &nbsp; 6&nbsp; | &nbsp; 5&nbsp; | &nbsp; 1&nbsp;| &nbsp; 2&nbsp;| &nbsp; 3&nbsp;| &nbsp; 4&nbsp;
}}
| {{mvar|B}}
{{aligned table|cols=8|class=wikitable
| &nbsp; 5&nbsp; | &nbsp; 4&nbsp; | &nbsp; 5&nbsp; | &nbsp; 4&nbsp; | &nbsp; 5&nbsp; | &nbsp; 4&nbsp; | &nbsp; 5&nbsp; | &nbsp; 4&nbsp;
| &nbsp; 6&nbsp; | &nbsp; 3&nbsp; | &nbsp; 6&nbsp; | &nbsp; 3&nbsp; | &nbsp; 6&nbsp; | &nbsp; 3&nbsp; | &nbsp; 6&nbsp; | &nbsp; 3&nbsp;
| &nbsp; 7&nbsp; | &nbsp; 2&nbsp; | &nbsp; 7&nbsp; | &nbsp; 2&nbsp; | &nbsp; 7&nbsp; | &nbsp; 2&nbsp; | &nbsp; 7&nbsp; | &nbsp; 2&nbsp;
| &nbsp; 8&nbsp; | &nbsp; 1&nbsp; | &nbsp; 8&nbsp; | &nbsp; 1&nbsp; | &nbsp; 8&nbsp; | &nbsp; 1&nbsp; | &nbsp; 8&nbsp; | &nbsp; 1&nbsp;
| &nbsp; 4&nbsp; | &nbsp; 5&nbsp; | &nbsp; 4&nbsp; | &nbsp; 5&nbsp; | &nbsp; 4&nbsp; | &nbsp; 5&nbsp; | &nbsp; 4&nbsp; | &nbsp; 5&nbsp;
| &nbsp; 3&nbsp; | &nbsp; 6&nbsp; | &nbsp; 3&nbsp; | &nbsp; 6&nbsp; | &nbsp; 3&nbsp; | &nbsp; 6&nbsp; | &nbsp; 3&nbsp; | &nbsp; 6&nbsp;
| &nbsp; 2&nbsp; | &nbsp; 7&nbsp; | &nbsp; 2&nbsp; | &nbsp; 7&nbsp; | &nbsp; 2&nbsp; | &nbsp; 7&nbsp; | &nbsp; 2&nbsp; | &nbsp; 7&nbsp;
| &nbsp; 1&nbsp; | &nbsp; 8&nbsp; | &nbsp; 1&nbsp; | &nbsp; 8&nbsp; | &nbsp; 1&nbsp; | &nbsp; 8&nbsp; | &nbsp; 1&nbsp; | &nbsp; 8&nbsp;
}}
}}

| Build the final square by multiplying the second square by <math>4n</math>, adding the first square and subtract <math>4n</math> in each cell of the square.
Example: <math>A + 4nB - 4nC</math>, where {{mvar|C}} is the magic square with all cells as 1.
{{aligned table|cols=8|class=wikitable
| &nbsp;33&nbsp; | &nbsp;26&nbsp; | &nbsp;35&nbsp; | &nbsp;28&nbsp; | &nbsp;40&nbsp;| &nbsp;31&nbsp;| &nbsp;38&nbsp;| &nbsp;29&nbsp;
| &nbsp;48&nbsp; | &nbsp;23&nbsp; | &nbsp;46&nbsp; | &nbsp;21&nbsp; | &nbsp;41&nbsp;| &nbsp;18&nbsp;| &nbsp;43&nbsp;| &nbsp;20&nbsp;
| &nbsp;49&nbsp; | &nbsp;10&nbsp; | &nbsp;51&nbsp; | &nbsp;12&nbsp; | &nbsp;56&nbsp;| &nbsp;15&nbsp;| &nbsp;54&nbsp;| &nbsp;13&nbsp;
| &nbsp;64&nbsp; | &nbsp; 7&nbsp; | &nbsp;62&nbsp; | &nbsp; 5&nbsp; | &nbsp;57&nbsp;| &nbsp; 2&nbsp;| &nbsp;59&nbsp;| &nbsp; 4&nbsp;
| &nbsp;25&nbsp; | &nbsp;34&nbsp; | &nbsp;27&nbsp; | &nbsp;36&nbsp; | &nbsp;32&nbsp;| &nbsp;39&nbsp;| &nbsp;30&nbsp;| &nbsp;37&nbsp;
| &nbsp;24&nbsp; | &nbsp;47&nbsp; | &nbsp;22&nbsp; | &nbsp;45&nbsp; | &nbsp;17&nbsp;| &nbsp;42&nbsp;| &nbsp;19&nbsp;| &nbsp;44&nbsp;
| &nbsp; 9&nbsp; | &nbsp;50&nbsp; | &nbsp;11&nbsp; | &nbsp;52&nbsp; | &nbsp;16&nbsp;| &nbsp;55&nbsp;| &nbsp;14&nbsp;| &nbsp;53&nbsp;
| &nbsp; 8&nbsp; | &nbsp;63&nbsp; | &nbsp; 6&nbsp; | &nbsp;61&nbsp; | &nbsp; 1&nbsp;| &nbsp;58&nbsp;| &nbsp; 3&nbsp;| &nbsp;60&nbsp;
}}
}}
If we build a <math>4n \times 4n</math> pandiagonal magic square with this algorithm then every <math>2 \times 2</math> square in the <math>4n \times 4n</math> square will have the same sum. Therefore, many symmetric patterns of <math>4n</math> cells have the same sum as any row and any column of the <math>4n \times 4n</math> square. Especially each <math>2n \times 2</math> and each <math>2 \times 2n</math> rectangle will have the same sum as any row and any column of the <math>4n \times 4n</math> square. The <math>4n \times 4n</math> square is also a [[most-perfect magic square]].