The world we live in is composed of many apparently consistent processes, which we attempt to discover. A key tool of discovery is mathematics. Mathematics is a language whose foundation is extremely rigorous and consistent. It is a tool of scientists, engineers, doctors, and philosophers. Computers can process mathematical procedures with amazing speed and can be used to produce programs (or procedures) to simulate real world processes. During the process of designing programs
(sometimes called algorithms) the analyst
Understanding the procedure will not directly help you solve the 4x4 matrix however basic concepts can be discovered by playing with this simpler problem on paper and in your mind. Later on we can address writing a program (i.e. committing the process we discovered to a computer program). That is not to say we can not use a computer as a tool of our thinking process. Used properly, computers are more reliable than your brain (at least mine) in remembering what has transpired in the past. Assuming you have discovered the process to generate "the basic 3x3 magic square", and thinking outside the box, low and behold you find the process can generate magic squares with sides whose length is a positive odd number (e.g. 5x5, 11x11, etc.). You will also discover this process does not apply to generating even numbered (4x4, 8x8, etc.) magic squares. Next you will find our basic 3x3 magic square, if viewed from a different angle (in our 3 dimensional world) has seven more perspectives. For example, take 9 children's blocks, the first block representing the number 1, the second block representing the number 2, etc. Place them on a glass top table to form the basic 3x3 magic square. When we look at the magic square from each side of the table we see four views of the same magic square. Now do the same looking at the underside of the table. Considering all rotations you should come up with a total of eight unique views of the "basic 3x3 magic square" illustrated below.
Problem 1: Can you create a 9x9 magic square (with integer numbers 1 though 81other than 1,2,...,9)? Problem 2: What is the maximum number of
odd sided magic squares that can be generated? Problem 4: Here is one solution to a 4x4 magic square. Can you come up with a procedure for generating other even sided magic squares?
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Charles Bovaird retired from IBM. He is presently Treasurer of DACS. You may contact him at treasurer@dacs.org. |