3 x 3 Magic Square

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Grade Vs Difficulty:

This is a circle that takes 1 to 2 sessions but can be extended to 4 or 5.

The starting question is: How many solutions are there to the 3x3 magic square? Perhaps some of the kids will know what one is but if not, they should ask enough questions to clarify what a magic square is.

(A 3x3 magic square is a 3 by 3 grid where the numbers 1-9 are exhaustively and only once each placed. All rows, columns and diagonals must add up to 15.)

Typically the circle will solve it once before talking about how many solutions there are. But, there will probably be guesses along the way.

Problem solving strategies I have seen:
1) Odds and evens can help simplify the problem. 15 is odd, odd plus even is odd and so on. There are five odd numbers and four even numbers…

2) There needs to be a balance between high and low numbers in a column or row or diagonal.

3) It is fairly easy to argue that the numbers 1-4 are too low for the center square and 6-9 are too high.

Once it is solved then the discussion begins in earnest about how many solutions there are. Essentially we have the symmetries of a square (or not!) This leads to a nice discussion of is there one solution or 8 solutions or other possibilities.

Created by Craig Daniels