MASC 380 Assignment #6

A magic square is a square array of positive integers such that the sum of the numbers in each row, each column, and both main diagonals is the same. (This sum is sometimes called the magic constant.) For example, consider the following 4 by 4 magic square, which appears in a 1514 engraving by Albrecht Dürer:

16	3	2	13
5	10	11	8
9	6	7	12
4	15	14	1

1. What is the name of the engraving in which this magic square appears? [1]

2. What contemporaries and predecessors of Dürer's constructed magic squares? [2]

There is a very nice algorithm for generating an n by n magic square when n is a multiple of 4:

• Write out the numbers 1, 2, ..., n² in order in a square array, starting in the top left-hand corner.
• Divide this array up into its 4 by 4 component blocks.
• In every position on a main diagonal of a component block, replace k by n² + 1 - k.

3. Execute this algorithm for the case n = 4. How is the resulting square related to Dürer's? [1]

4. Show that the algorithm gives a square which is symmetric: if a_ij denotes the number in row i and column j, then a_ij + a_st = n² + 1 whenever a_ij and a_st are in symmetric positions relative to the centre of the square (i.e. whenever i + s = j + t = n+1). [1.5]

5. Prove that the algorithm does give a magic square. What is its magic constant? [3]

6. What additional properties do symmetric magic squares have? [1.5]

Department of Mathematics

Trent University