A Square of Circles

The Problem

Using the numbers 1 – 9 put a different number in each circle. Arrange the numbers so that the sums of the three numbers on each side are
 

• all different sums
• all the same sums
• all even sums
• all odd sums

What is the biggest and smallest side sums that you can make?

Teaching sequence

1. Introduce the problem to the class – you could do this by asking a student to select a digit label from a "hat" and record the number in one of the circles. Ask another student to select a number and record it. Continue until the circles are full and then ask the students to make statements about it.
2. Read the problem with the students – make sure that they understand that they are to use each of the numbers 1- 9, once only.
3. Get the students to work on the problem individually or in pairs. Encourage the students to look for patterns in the combination of odd and even numbers as they solve parts c and d.
4. Share answers to each of the parts of the problems. Ask the students to make statements about the combination of odd and even numbers.

Extension to the problem

Arrange numbers that that all the rows and columns have the same sum.  How many ways can you do this?

Solution

There are many possible answers to the first two parts of the problem.  We give some of these below.  One way to do the first one might be to try 1 in the top left corner, 2 in the next column to it and 3 in the top right corner.  If we keep putting the numbers in systematically, intuition tells us that the top and bottom rows will be different.
1     2    3  (6)
4     5    6 
7     8    9  (24)
(12)    (18)

To get all the sides the same, we could try to put the small numbers in the corners and balance them up with the larger numbers.
1     8     3
9     7     5
2     6     4
That may take a bit of experimenting.  You need to make sure that 1 and 4 are not on the same side.

Both examples we've used so far have even sums on the side. To find some odd sums we'll put the small numbers in the corners but in order this time.
1   6   2
5   9   7
3   8   4
Once we've put in 1, 2, 3 and 4 its just a matter of choosing an odd number in the sides where the sum is already even and an even number where the sum is already odd.

Note:

The only ways to get an odd row are to sum 2 evens and an odd or 3 odds.

• even odd even
• odd odd odd
• even odd even