Magic Squares

The 'magic square' is a model with inputs in the grid of squares and outputs in the totals for the rows, columns and diagonals. The rules for one type of magic square:

• for squares with an odd number of values each total should equal the product of three times the central value e.g. 5:15;
• the numbers around the central value should be evenly distributed around it e.g. 5: 1-4 and 6-9;
• a number may be used once only;
• the total in every output square must be identical.

Magic squares can be solved on paper or in your head but they can also be solved with a spreadsheet.

The magic square model is set up as follows:

1. Define a grid 3 cells x 3 cells in B3:D5 (you can try 4x4 and 5x5 later)
2. Select cell B6 and use the S button to get the B column total
3. Drag the copy marker from B6 through C6 and D6
4. Select cell E3 and use the S button to get the Row 3 total
5. Drag the copy marker from E3 through E4 to E5
6. Select cell E6 and enter '=B3+C4+D5' to get the first diagonal total (use the mouse to select the cells)
7. Select cell A6 and enter '=B5+C4+D3' to get the second diagonal total.
8. Format the cells to make a square

You can now enter values to solve a magic square puzzle, for example, use digits 1-9 to make each total equal to 15. You could start with some values which point the way or, for a stiffer challenge, start with an empty grid. You should find it easier to solve the problem with a spreadsheet, changing numbers around, following a strategy, etc. than with paper and pencil. 

Each time you enter a new number the totals are updated automatically so you can see the effects across the whole model of one small change. If the outputs were measurements of wind resistance on the body of a car you could judge the effects of small changes in design on the performance of the whole car. Another example Once the model is set up you can change the values at will and observe the effects. The main difficulty in modelling is finding suitable equations for the process you want to model.

There is a method you can use to enter the values into a magic square. Clear the values in the yellow input squares and follow these steps:

• Starting with an empty square place the starting value, e.g. 1, in the top row of the middle column.
• Place the next value, 2, at the bottom of the next column (rule: at the top of a column move to the bottom of the next column)
• Place the next value, 3, in the first column of the row above (rule: at the end of a row move to the first column of the row above)
• Place the next value, 4, underneath the 3 (rule: if the next cell one column to the right and one row up is occupied drop to the cell below in the current column)
• Place the next values, 5 and 6, along the central diagonal (rule: if the cell one row up and one column to the right is empty place the next value there)
• Place the next value, 7, in the cell below (previous rule: if the cell one row up and one column right is occupied drop one in the current cell)
• Place the next value, 8, in the cell in the first column one row up (previous rule: at the end of a row move to the first column of the row above)
• Place the last value, 9, in the cell at the bottom of the next column (previous rule: at the top of a column move to the bottom of the next row)