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Inside Triangles

Stage: 1 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The Four Mathmateers at Brocks Hill Primary School said they drew lots of triangles and used trial and error (or trial and improvement) to answer the question. I think, though, that they might have counted the same triangle more than once.

Starting off using trial and improvement is an excellent idea! How will you then decide whether you have found all the triangles?  Try looking at the hints in "Getting Started" for some ideas.

Neptune Class from Riverley Primary wrote to say:

We think that there are $6$ triangles in total.

We made sure that each triangle was a different type (scalene, isosceles, right-angled and equilateral) and we experimented with different shapes on the pin-board.

We thought that there must also be a mathematical way to systematically calculate the solution to the problem, but we're still working on it.

I think there are a few more than six triangles, but I like the way you thought that working systematically would help.  That's exactly what Greg from Swanland County Primary School did.  He said:

First I tried to just make triangles with a dot inside.

I found these but I thought there might be more.

So I tried every one with a three dot base.


Then I tried ones with a four dot base.


But the purple one here is the same as the green one above. So only $2$.

Then I tried one with a two dot base.

I think there are $7$ different triangles.

Swifts Class from Southill Lower School said that they found $9$ triangles with one dot but unfortunately they didn't send us a picture of their triangles.

I think I agree with Swifts Class.  Can you find the two triangles that Greg missed?  Let us know if you do.

We had a solution come in from Lily-Mai and Dillon from Millbrook School in Swindon. Their teacher wrote to say that they think they have found all nine triangles, and would love to see their solution on the NRICH website! Lilly and Dillon worked systematically, finding all triangles with a base of 2, then 3, then 4.

Yes, all 9 are here, well done.