Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

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?
What do you notice about all the solutions you find?

Can you explain what you see?

Can you convince someone that you have all the solutions?

What happens if we use the numbers from 2 to 6? From 12 to 16? From 37 to 41? From 103 to 107?

What can you discover about a V that has arms of length four using the numbers 1 to 7?

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.

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

You can see Lynne McClure introducing Magic Vs to a small group of teachers on YouTube here and she takes the ideas a bit further in this clip.

You could start by having two Vs displayed on the board (interactively if possible), one which is "magic" (i.e. whose arms have the same total) and one which is not. Ask the children to talk about what they see. If it doesn't come up naturally, draw their attention to the total of each arm and introduce the term "magic V".

Ask learners to suggest some questions they could ask about magic Vs and then direct their attention to finding out how many other magic Vs there are, using the numbers $1-5$. Children could work in pairs, using digit cards to try out their ideas. They could record magic Vs on this sheet . After a some time,
bring them together to share some of the magic Vs they have found so far. You could record them on the board and then invite learners to comment on what they notice. Can they offer explanations?

Having given them more time to explore and make generalisations, you could allow learners to pursue one of the questions they asked at the start of the lesson. These might include, for example, investigating different ranges of numbers or Vs which have four numbers in each arm.

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

Children can be challenged to investigate Vs of different sizes with different ranges of numbers. Are they always possible to solve and can they predict the number of solutions? Further extensions to this problem include other arrangements of the numbers, for example a magic cross, or the use of negative numbers, or how about a multiplication Magic V?

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.