### 3x3 (not pure) in 5x5 (pure) magic square

Use the row and column grids of the 3x3 magic square to construct the 3x3 inlay.

 1 2 0 0 2 1 0 1 2 2 1 0 2 0 1 1 0 2

To get the right numbers of the 3x3 inlay add 1 to all numbers.

Row grid 3x3 inlay                                    Column grid 3x3 inlay

 2 3 1 1 3 2 1 2 3 3 2 1 3 1 2 2 1 3

Construct the row grid of the 5x5 border and take care that the sum of opposite numbers in the border is allways 4 and the sum of a row or a column is allways 10. Put the numbers as follows (n.b.: Put 2x the middle number from 0 up to 4 cross in the corners):

Row grid 5x5 border

 2 1 3 4 0 0 4 4 0 0 4 4 3 1 0 2

Construct the column grid of the 5x5 border and take care that all combinations of row coordinates and column coordinates are unique, so you get all the numbers from 1 up to 25 in the magic square.

Column grid 5x5 border

 0 4 4 0 2 1 3 4 0 3 1 2 0 0 4 4

Take 1x a number from the row grid, add 5x the number of the same cell from the column grid and add 1.

1x number from row grid          +        5x number from column grid + 1 =     3x3 in 5x5 magic square

 2 1 3 4 0 0 4 4 0 2 3 22 24 5 11 0 2 3 1 4 1 1 3 2 3 6 8 19 12 20 4 1 2 3 0 4 3 2 1 0 25 17 13 9 1 0 3 1 2 4 3 2 1 3 1 16 14 7 18 10 4 3 1 0 2 2 0 0 4 4 15 4 2 21 23

Use this method to construct inlaid squares of odd order from 5x5 to infinity. See

5x5, 3x3 in 5x5.xlsx