# Term 3 - July 30th (3-6)

##### Week beginning July 30th – Year 3 to Year 6

Magic Vs

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 4 using the numbers 1−7?

#### Solution

If we have the numbers 1, 2, 3, 4 and 5, and our goal is to arrange all 5 numbers in a way such that the 2 ‘arms’ have the same total. If we get the total of all the given numbers, we get 15. In the middle, we must have an odd number, because if we have an even, say 2, 15-2=13 and 13 is not divisible among the other 2 circles of arms. If we pick 1 as the middle, then each side is required to have a total of 7 on the other 2 circles. For 3, we need a total of 6 on the 2 circles, and for 5, we need a total of 5 on the other 2 circles. We have 24 different permutations. Here is the formula:

Slot 1: 4 possibilities
Slot 2: 1 possibility
Slot 3: 2 possibilities
Slot 4: 1 possibility
Middle number: 3 possibilities (3 odd numbers between 1 to 5, inclusive)
So 3*(4*1*2*1) = 24.

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