Maths Week Scotland 2023 - Problem 5 - Triangles

CHRIS: This challenge is all about triangles and figuring out how they fit together.

Let's take a look.

Here's a diagram of a large triangle made up of lots of smaller triangles. Can you work out how many triangles there are in this figure altogether?

Explain your solution in detail.

What are you starting with Kayla? Luke is going to start with all the wee triangles because they're the obvious ones.

PUPIL: As well as the small triangle and the big triangle, there is lots of other sizes to count.

CHRIS: So how many like that can you find in there? There must be a few.

PUPIL: It might help to add up each size of the triangles separately.

PUPIL: Remember that the triangles might point down instead of up.

Do upside down count just as well as, like, upright ones?

CHRIS: Yeah. Perfect, yes. You need to include all of them. Any triangle that you can see.

PUPIL: Give it your best shot.

This triangle challenge was tricky. Let's look at the solution together.

The first important thing to notice about this diagram is that there are triangles of different sizes.

There are the individual small triangles. But there are also triangles made up of four small triangles together; nine small triangles together; sixteen small triangles together; twenty five small triangles together; and the big triangle is made of thirty six small triangles together and there are triangles that point down as well as triangles that point up.

To start with you could count small upright triangles inside the diagram. There were twenty one of these.

Fifteen upright triangles made up of four smaller ones.

Ten upright triangles made up of nine smaller ones.

Six upright triangles made up of sixteen smaller ones.

Three upright triangles made up of twenty five smaller ones.

And finally, the whole diagram is one big triangle made up of thirty six smaller triangles.

Twenty one, we're already at fifty seven just from those two, but obviously it gets less as we go round, yeah?

Now we need to count all the upside down triangles. There are fifteen upside down small triangles.

There are six upside down triangles made up of four smaller ones.

And there was one upside down triangle made up of nine smaller ones. So let's add up all our triangles together now. For upright triangles, we found twenty one plus fifteen plus ten plus six plus three plus one, which is fifty six. And for upside down triangles we found fifteen, plus six plus one, which is twenty two.

Fifty six plus twenty two equals seventy eight.

There were seventy eight triangles in this diagram altogether.

Great work.