Concentric Magic Squares
'Take magic squares to a new level and program a spreadsheet to help you!'
Magic squares are fun, simple puzzles where numbers are placed in a square arrangement according to a given rule. In this case, we start with - 'Can you arrange the numbers 1 to 9 in a square such the the total of all the numbers in each of the columns, each of the rows and both the diagonals is the same?' Can you do this with any set of 9 consecutive numbers? What are the tactics? Where is the best place to start? How are the groups of three related? What should the total of the numbers be? Exploring these questions and the more general relations in this problem is the key to solving the concentric magic squares problem - a magic square, within a magic square, within a magic square. Equally essential is being able to program a spreadsheet to do calculations for you as the size of the square increases!
The problem is outlined in the Concentric magic squares worksheet. This can be used on paper or electronically since you will need access to a spreadsheet as well.
The following is a brief screencast to explain the idea of magic squares and help with entering simple formulae.
The following is a brief overview of the activity
- Arrange the numbers 1 to 9 in a magic square as described above.
- Program a spreadsheet to do the calculations so that the user is only required to move the numbers around.
- Explore any set of 9 consecutive numbers. Try with the numbers 9 to 17.
- Use the answers to the above when programming a new grid to arrange the numbers 1 to 25 in a magic square, such that the 3 x 3 square in the middle is also magic!
- Now the big one! Work with the numbers 1 to 49 to make a 7 x 7 square with a 5 x 5 square inside it and 3 x 3 inside that - all magic.
- Throughout this exercise, the object is to observe and use the patterns and relations from the previous step to help structure the next one. Each solution should inform the next.