10 x 10 Magic Squares

The technique which produced the The 6x6 Magic Squares provides the basis for making larger 4p+2 magic squares. For the 6x6 square, a 3x3 square was doubled in size and then combined with a Magic Grid. For a 10x10 square a 5x5 magic square is "doubled" in size in the same fashion.

Using the 5x5 Square as a Basis.

The The 5x5 Magic Carpet used on the page about 5x5 magic squares is shown here. The square is doubled using the same technique that was applied to the 3x3 to make the The 6x6 Magic Square. Each cell is duplicated in both axes to produce the square on the right.

This square is used twice, once rotated through ninety degrees. One of the squares is multiplied by 20 and the other by 4. The two are added together in preparation for being added to the Magic Grid.

The Magic Grid for this 10x10 Square.

The Basic Pattern to go with this square has to be lengthened to fit this new larger size. The one used for the 6x6 is merely elongated. A single zero is followed by N/2 ones with the remainder of the row filled by zeros. The next line is the inverse. These two rows are repeated down the square. The final pair of rows, otherwise still the same, is reversed left for right. This pattern is used once for one axis and then, as in the case of the 6x6, rotated through a right angle for the other axis.

   Basic Pattern        0 1 1 1 1 1 0 0 0 0     1 0 1 0 1 0 1 0 1 0
                        1 0 0 0 0 0 1 1 1 1     1 0 0 1 0 1 0 1 0 1
0 1 1 1 1 1 0 0 0 0     0 1 1 1 1 1 0 0 0 0     1 0 0 1 0 1 0 1 0 1
1 0 0 0 0 0 1 1 1 1     1 0 0 0 0 0 1 1 1 1     1 0 0 1 0 1 0 1 0 1
                        0 1 1 1 1 1 0 0 0 0     0 1 0 1 0 1 0 1 0 1
                        1 0 0 0 0 0 1 1 1 1     0 1 0 1 0 1 0 1 0 1
                        0 1 1 1 1 1 0 0 0 0     0 1 1 0 1 0 1 0 1 0
                        1 0 0 0 0 0 1 1 1 1     0 1 1 0 1 0 1 0 1 0
                        0 0 0 0 1 1 1 1 1 0     0 1 1 0 1 0 1 0 1 0
                        1 1 1 1 0 0 0 0 0 1     1 0 1 0 1 0 1 0 1 0

Properties.

These Magic Grids are designed so that, when the values in the first square are doubled and added to the values in the second, the result is an array of 2x2 squares (on the right), each one containing the four numbers 0, 1, 2, and 3. The magic property for the whole square is retained.

The 10x10 Magic Square.

The numerals in the array of small squares are added to the double size 5x5 square above to make the final 10x10 magic square (on the left below).

Variations.

Just as with the 6x6 square, there are four component magic carpets: two expanded 5x5 squares and two magic grids. These components can be used to make a whole family of related but distinct magic squares. The four component squares described above are multiplied by 20, 4, 2, and 1, Alternative sequences can be used and will make squares which look very different, e.g., 4, 20, 2, and 1; or 1, 5, 25, and 50. The squares produced by this strategy appear to be so different that their common parentage can only be recognised with appropriate Analysis Tools.

Conclusion

This page descibes an orderly, logical system for making magic squares of order 4p+2. It also explains how squares with a strikingly different appearance may share an underlying family of magic carpets as parents. Although there are other patterns which can be used in conjunction with an expanded odd-order square, the advantage of the Basic Pattern described here is that it is readily visualized and is easily expanded for higher orders of the 4p+2 series of magic squares.

The 10x10 Magic Squares