Magic Triangle

'Magic Triangle' printed from https://nrich.maths.org/

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Magic Triangle

Place the digits 1 to 9 into the circles so that each side of the triangle adds to the same total.

Can you find more than one solution?

Why do this problem?

This is a 'low threshold high ceiling' problem in that all children will be able to find at least one solution through trial and improvement, but others will use logical thinking to develop and try out different strategies for making the sides equivalent.

Possible approach

There are different layers to this activity, making it ideal for a whole class task.

Begin by showing the class a large version of this task on the board and write the numbers 1-9 at the side. Put numbers randomly in the circles, crossing them off the 1-9 list as you do so. When all the circles are full, add up each side of the triangle and record the answer. Ask the children if they think it's possible to get the answers closer than yours. Give them some time to explore, using whiteboards or the downloadable sheet, and see if they can improve on your result.

When someone gets all three sides the same, stop the class and record the result on the board. Ask if they think this is the only way we could place the numbers to make the sides equivalent. Allow some time for exploration and if possible make a part of the wall space available for recording all the children's answers. If you encourage the children to record each solution on a separate piece of paper you can rearrange them during discussion.

Ask children if they think trial and improvement is the only way of finding solutions. Some of the 'key questions' below might be helpful to prompt children to think more logically about the possibilities for each side.

You might choose to leave this as a 'simmering' activity over a period of a few days. Not all children will engage with it but some will become very enthusiastic. In the plenary discussion draw attention to families of solutions, and the upper and lower limits for the side totals.

Key questions

What's the biggest total a side could have? How do you know?

What's the smallest? How do you know?

Does thinking about odds and evens help?

What do all of the nine numbers add up to? Is this information helpful?

Try some different possibilities for the numbers at the three vertices. What do you notice?

Possible extension

Encourage the children to tweak this question by asking 'what if ...?' Suggestions you could throw in might include:

What if it was a square not a triangle?

What if we had five numbers on each side and we filled the triangle with the numbers 1 to 12?

What if we only used the first nine odd numbers? What if we only used even numbers? Or the first nine multiples of ten?

What if ...?

Providing a public space for the children to record their own investigations makes it a more collaborative activity.

Children could also try a more challenging version of the problem.

Possible support

Children who find this difficult could begin with an easier activity using six circles, three on each side, and the numbers 1-6. Begin by giving them 15 buttons or counters to place in the circles so that there are the same number of counters on each side. This gives visual support. Move from there into having a different number of counters in each circle, and from there to using digits rather than counters.