## 14 Mar 2013

### All the Grids of Sums: Upper Secondary Mathematics Competition Question

The diagram below is the same as that used in the Grid of Sums question. However, in this case, I would like you to find all the possible solutions.

To restate the rules, the number in each circle is the sum of the numbers in the two squares above it. You must use every number from 1 to 9 only once.

List all the possible solutions. How many unique solutions are there? You may ignore reflections.

You will come across many questions such as this one, especially in any follow-up papers to open maths competitions. At first glance, there seem to be a huge number of options. Putting the nine numbers in randomly, there are 9!/2 permutations – too many to enumerate. However, the grid has some restrictions that will help you reduce the number of options dramatically.

This question does require enumeration, but you need to find a systematic way of doing it so that the answers can be found within a reasonable time. It also means that you can be confident of having found all the solutions.

Level: Upper Secondary (Red)