ALMOST MAGIC SQUARES

A New Twist to a Familiar Problem

Have you ever observed a student who when presented with a magic square and instructions to verify that all the rows, columns, and main diagonals have the same sum, finds the sum for a few of those sets and then jumps to the conclusion that all the remaining sets have the same sum? As the student does not carry out all the work, much of the intended drill is lost, and likewise any self-checking aspects. This article describes some easily constructed lesson materials that prevent such an assumption being made.

The concept is deceptively simple: fix one of the entries in the square so that it is "incorrect". The object of the exercise then is to find the "culprit"; determine by how much it should be altered to bring things back into balance; and make the correction.

Here is a simple example that can be used to demonstrate the idea to your class. Note that the second row and third column each have sums that are one greater than the other rows and columns. (This is why I call such squares almost magic.)

 11 4 9 24 6 8 11 25 7 12 5 24 24 24 25 11

Now, if the entry that is common to both sets (11)is decreased by one, the magic property of the square is easily restored.

To construct a variety of almost-magic squares, follow these steps:

1. Select a magic square of the size you desire.
2. Add one value to all the entries, except one; add a different value to that final entry.

Before performing Step 2, it is often helpful to multiply all the square's entries by a particular value. This increases the variety of possible problems, and more importantly, allows you, as the teacher, to adjust the difficulty level of the addends.

The following are some examples that I have used in my teaching of elementary school students. There is sufficient drill work in any one square and each provides a needed experience in problem solving. These exercises would make good worksheets to leave for a day when a substitute is needed.

ONE TOO BIG
 4 14 12 19 10 2 8 6 16
 8 18 5 6 10 14 16 2 12
 6 21 18 27 15 3 12 10 24
 7 22 19 28 17 4 13 10 25
 9 29 25 37 21 6 17 13 33
 8 28 24 36 20 4 16 12 33

TWO TOO BIG
 11 38 31 46 26 6 21 16 41
 35 7 27 15 23 31 21 39 11
 16 2 3 13 5 13 10 8 9 7 6 12 4 14 15 1
 32 4 6 26 10 22 20 16 18 14 12 24 8 28 32 2
 17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 21 2 9

SOME TOO BIG
 19 5 6 16 8 14 13 11 12 10 9 15 7 20 18 4
 17 3 4 14 6 12 11 9 10 8 7 19 5 15 16 2
 48 6 9 39 15 33 20 24 27 21 18 36 12 42 45 3
 52 32 48 4 12 40 24 60 8 44 21 56 64 20 36 16

```This article appeared in The Oregon Mathematics Teacher, October 1978.

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