3-2-1 Totals

As it only uses the numbers 3, 2 and 1 to make totals, this problem is very accessible but equally has the potential for lots of mathematical thinking.

It is another problem created that could be used for the theme of ‘Three is a magic number’ for Maths Week England 2023. Find out more about that here and see the other tasks I have planned on the theme of three here.

Here we have a grid of four triangles with the numbers 3, 2 and 1 inside them:

If we combine the triangles into different regions, and each region had to have a different total, what would be the lowest possible value of the region with the highest total?

In this example, it is fairly trivial to find the solution. The yellow region has a total of five. I cannot split up the regions in another way that results in a lower highest region total.

Combining the one and the two would make three, but that would leave two threes in their own region. Therefore, we have to combine the three and the two to make five and create two other regions of three and one.

What if we had a bigger grid?

Adding an extra row makes it a little more challenging. Each row follows the repeating pattern of 3, 2 and 1.

In this case, we can keep the region with the greatest total down to just six:

It is interesting to note though that we have not created a region with a total of one. This seems illogical but with only two ones to play with, they were useful in offsetting the red and green regions from other regions. It may be that the children start by creating a region of one, then two, then three as that seems like a systematic approach. However, if you try it, you will discover that it leads to the wrong number of triangles left over.

Having had totals of five and six, it will inevitably lead to the conjecture that if we add another row, the region with the greatest total will be seven. However, you might equally get them to consider why this may not be the case. They can do a lot of thinking before they even attempt it.

In the row we have added, the numbers total 15. Given that we had regions with values from two to six in the previous grid, even if we had a region that totalled one as well, we could not just add a region that totalled seven to compensate for the extra 15.

Therefore, at this point, there is going to be a much wider variety of attempts from the children. That is great and they can think about what strategies they are using. Which parts of the triangular grid do they prioritise first? Since the row we are adding has a total of fifteen, two regions of seven and eight should be enough. I started with those:

There is something satisfying about this solution that happened completely by accident. That’s maths for you sometimes! The splitting of regions is symmetrical. This could be another avenue to explore. What if the grid had to be symmetrical? I went back to the previous grid and came up with this:

Notice the similarities and differences between the two symmetrical solutions.

Going back to the original set of constraints, we can continue adding rows:

The bottom row totals 18. We can once again think systematically about our strategies and try and create regions of ten and nine first. It is perhaps at this point, if we have this strategy, that further exploration becomes less worthwhile. Time to look at alternative ideas!

I have already introduced one idea, the grid needing to be symmetrical but there are so many other possibilities too.

What if we changed the order of the numbers?

We could instead have the rows go one, two, three rather than three, two, one. Or you could have two, three, one or indeed many different patterns of numbers. Are the strategies still the same?

What if we tried to create regions that all had the same value?

What would be the smallest value possible? Or if it is not possible to complete the grid only with regions of the same total, what would be the smallest total of the remaining cells?

What if we started with just no triangles and added triangles to make regions contain one then two, then three etc. up to ten? What would be the smallest perimeter possible?

I find this idea intriguing and is an example of the creativity that is possible from an initial idea. In this, a triangle with a one can never share a side with another triangle with a one, a two can never share a side with a two, nor a three with a three.

For example, I have placed the one, two and three below into single-triangle regions to make the numbers one, two and three.

Now, I need to make four.

I could add a region like this to make four. I have used a three and a one to do this. Next I need to make five. I could use a three and a two or a one, a two and a two or another way:

As mentioned before, the aim is to go all the way up to ten and use the smallest perimeter possible. It is a similar theme but a new problem. Once a culture of creativity is established, children will come up with ideas like this themselves.

Children’s possible use of reasoning skill

Search
Search inquiry reasoning skill in maths

As mentioned, it would be tempting to Comb by starting with a region of one and then two and then three. This seems systematic. However, it leaves triangles left over that cannot be optimally grouped. Instead, it is better to consider what might be the highest value region and go downwards.

The other way that children will Search is by the choice of triangles they use to build regions. Will they start in the corners, the middle or somewhat randomly?

Organise

An organised list of how combinations of three, two and one can be used to total particular numbers is useful when considering which triangles to use.

Discover

I mentioned one possible conjecture above. With each row, particularly with the strategy outlined above, children can conjecture as to what they think the maximum value will be and then use this to develop a strategy for each new case.

A more general strategy for n rows would be trickier to develop.

Another thing to explore and notice is the number of regions for each case.

Investigate
Investigate inquiry reasoning skill in maths

To get children away from working randomly, the highlighting of strategies and then investigating them is important.

Argue

I have highlighted above possible examples of how children might have to explain their thinking. There is a lot of scope here. It could be you go in a different direction after the first case and consider what numbers the next row should even contain.

Explore

This is an island that is easy to adapt into new problems. Children can change the numbers, the shape, add constraints, all sorts. I’d love to hear about their ideas.

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