Triangular Threes

Maths Week England in 2023 has the theme ‘Three is a Magic Number’.

I came up with this task with the intention of using at my school for MWE. It involves creating an ever-growing web of calculations inside triangles that result in the answer three. A lot of thinking and trial and error is required but the webs end up being much bigger than you might initially think.

I have also planned other tasks using the theme of ‘Three is a Magic Number’ that could be used during MWE which can be found here and many more open-ended investigations for a range of curriculum objectives that can be found here.

Let us start with the numbers one, two and three arranged around a triangle. If you want an extra challenge, start with zero, one and two instead.

Can we make a calculation using all three numbers with an answer of three?

Now let’s add in a four. It has to connect to two of the numbers to form a new triangle. Those three numbers together must be used to form a calculation with an answer of three. Like this:

Then we need to add five. I have multiple options as to where it could go but I have placed it here:

We then add six, seven, eight… How many numbers is it possible to add to our web?

Below I have got nine numbers. When I added the ninth number, it created two new triangles and I was required to find calculations that resulted in three for both.

Here is a continuation up to 10:

The nice thing is that it is likely that everyone that attempted this would have a unique web as there are so many different possibilities. It is definitely possible to reach a number much higher than ten but how high can you go? I would love to hear what you come up with!

Some webs do seem to become impossible to go further with. The children can consider what the best strategy is in terms of the shape of the web. Is it better to create a circular or spikey shape for instance?

What if we started from zero?

As mentioned previously, this makes for a more challenging start as zero is not very helpful with many pairs of numbers to reach three. It is possible though.

What if we had four numbers around squares?

I have not tried this so I am not sure whether or not it would be easier or harder. In some ways, there are more numbers to work with so it will be easier but equally, there are then more numbers that need including in a calculation.

Children’s possible use of reasoning skills:

Search inquiry reasoning skill in maths

This island focuses on Seeking. We need to analyse the web, try out different positions and think about where is best to place the next number when more than one possibility exists. This can involve looking ahead and ensuring that we have an idea of how connections could be continued.


The children are not likely to focus on using the Organising skills as much with this island. In a sense, the whole web is an organisation of numbers though.


Lots of strategies can be Discovered such as making multiples of three to divide or thinking about using the difference between two numbers effectively. Once the children get going, they will start to notice that similar strategies can be used continuously.

There is not going to be a strategy that works for every number but maybe they can get a feel for when different strategies are likely to work.

Investigate inquiry reasoning skill in maths

This will continuously be used as the children will build up a bank of strategies and then test them out with each new number.


There is a lot of potential for this. The children can think about why three numbers would or would not be a good grouping around a triangle. When a few numbers have been added, there are so many possible positions available to them. As such, they need to convince themselves or a partner about which positions to focus on. As such, this would make a great task to work on with a partner.


I have given the obvious ‘What if…?’ questions that might continue this exploration. However, this is an island that does not require lots of new directions to be effective. There is a lot of potential in the base task itself.

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