Magic Vs

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

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Magic Vs

The problem is explained below, but you may wish to scroll to the foot of the page to watch a video of the NRICH team presenting this challenge.

Place each of the numbers 1 to 5 in the V shape below so that the two arms of the V have the same total.

How many different possibilities are there?
How do you know you've got them all?

What do you notice about all the solutions you find?
Can you explain what you notice?

Can you convince someone that you have all the solutions?

What happens if we use the numbers from 2 to 6?
Can you explain what you notice this time?

Here is a video of the NRICH team presenting the challenge. You could just watch the start to check that you understand the problem, or you may like to pause the video and work on the task at various points.

Click here for a poster of this problem.

If you have enjoyed this problem, you might like to explore Magic Letters.

Why do this problem?

This problem gives opportunities for children to make conjectures, prove these conjectures and make generalisations. They will be practising addition and subtraction, and applying their knowledge of odd/even numbers.

Possible approach

In pairs, invite learners to arrange the numbers 1-5 in a V shape. They could use digit cards and/or record using this sheet. Ask them to add the three numbers in each arm. If the arms have the same total, explain that it is a ‘magic V’.

Allow a few minutes for pairs of learners to find some magic Vs. Ask one pair for their magic V and record it on the board for the whole class to see. Explain that you are going to call the sum of the three digits in an arm of a magic V the 'magic total'. What is the magic total of this V?

Challenge the class to find other magic Vs with this same magic total as the one on the board. How many can they find? How do they know they have found them all? As they work, look out for those pairs who have developed a system, or way of working, so that they know they don't miss out any possibilities.

After a suitable length of time, encourage a pair to explain how they found all the possibilities, and help the class to articulate how they know they have found them all. Explain to the group that, for this task, we'll consider the eight possibilities with the same magic total as being the same.

Invite pairs to use the numbers 1-5 to create magic Vs with other magic totals. Bring everyone together again and write up the new magic Vs. Ask children to think about what is the same and what is different about these magic Vs. They might mention that they all use 1-5, but in a different order. They might notice that all the magic Vs have an odd number at the bottom of the V. Challenge learners to explain why an odd number has to go at the bottom of a magic V. If they need a nudge, you could suggest that they find the total of the numbers 1-5. How might this help?

Can they predict what will happen with numbers 2-6? Can they explain their reasoning? Ask them to try it out - were they correct?

Key questions

What do your magic Vs have in common?
Can you explain why?
What would happen if we used five different consecutive numbers?
Can you explain why?

Possible extension

What are the magic totals for numbers 1-5? What are the magic totals for numbers 2-6? Challenge children to think about what they notice. Can they find a magic V that has a magic total of 100 (using five consecutive numbers)?

For a Christmas version of Magic Vs, see this PowerPoint File, which was created by Rachael Vaughan, a teacher at Bute House Prep School.

Possible support

Using mini-whiteboards and digit cards may 'free up' some children so that they don't worry about getting a magic V straight away.