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?

What is this problem about?
This type of problem provides the opportunity for the children to do a lot of calculations in an interesting situation. It also reinforces the notion of odd and even numbers and combinations of odd and even numbers. The children can be encouraged to notice that an odd + odd = even, even + even = even, odd + even = odd.

We expect that most children will tackle this by guessing and checking. There are sufficiently many possibilities for the answers to the first three parts of the questions to come out reasonably quickly.

Achievement Objectives
Number (Level 2)
- recall the basic addition facts
- mentally perform calculations involving addition

Mathematical Processes
- devise and use problem solving strategies (guess and check, act it out, draw a picture)
- use equipment appropriately

Resources required
digit labels (BSM equipment)
Blackline master of the problem (English)
Blackline master of the problem (Maaori)

Specific learning outcomes
The children will be able to:
- recall basic addition facts to 20
- investigate odd and even numbers

Teaching sequence

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

Extension 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 inorder 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.