Algebraic Pattern

Pandiagonal Magic Square of Order 4

Nearly all magic squares and cubes, which John R. Hendricks shows in his book Inlaid Magic Squares and Cubes base on algebraic pattern. This way a great variety of magic squares can be build with different solution sets.

Let me explain this method with an example. There are three different representations of a pandiagonal magic square of order four. One of them is

The letters a, b, A and B in this algebraic representation denote digits in the number system with base n. The use of small and capital letters denote complementary digits:

Now we have to build a solution set. There are four digits in the quaternary number system, so we may choose for example

This means that we have a pandiagonal magic square of order four in the quaternary number system. John R. Hendricks calls this an intermediate representation.

You can go now directly from the intermediate representation to the conventional representation in the decimal number system. The decimal number number can be calculated with

And here it is, one pandiagonal magic square of order four.

Pandiagonal Magic Square of Order 5

An algebraic pattern for a pandiagonal magic square of order five is easy.

Please notice with the pattern shown that capital letters were not required because with this pattern a pandiagonal magic square is achieved for all assignments of the digits 0,1,2,3,4 and will produce an intermediate representation of a magic square in the number system radix 5. That means that there are

possibilities to create such a pandiagonal magic square. I choose

which will create the following magic square:

Conclusion

All magic squares and cubes in his book Inlaid Magic Squares and Cubes are based on such algebraic pattern. With more than seventy different patterns you can create a great variety of different magic squares. You will find some of them as examples in the category Hendricks/Examples on this web site.