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Nine-pin Triangles

Age 7 to 11 Challenge Level:

Cong who goes to St. Peter's RC Primary, Aberdeen, sent in a correct solution to this problem. The key to answering it is to be sure you know what you mean by "different" triangles. Cong found 7 different triangles could be drawn on the nine-pin board which he drew:

nine different triangles

He also sent in a table which gave some more information about each triangle:

Number Colour Type
1 Green Isosceles
2 Light blue Scalene
3 Purple Scalene
4 Orange Isosceles
5 Pink Scalene
6 Blue Isosceles
7 Red Equilateral

Well answered, Cong, thank you.

Theodore from Abingdon Elementary School in the United States of America had a different idea of what 'different' means!  He wrote;

First, I realized that I had to come up with a theory that could solve the problem using basic math. Second, I used one of the dots as a base dot and made triangles off of that one using the dot to it's left and kept going down the circle dots. For the first dot it was seven. I then figured out that for the dot next to it (left) would be one less triangle than the one to the right of it. For that, I got 28 triangles. I then realized that I would have to do it for all the other dots using them as the base dot, accept subtracting one for each one down the line. I then added
28,27,26,25,24,23,22,21,and 20. My final answer was 216.

Richard from Cliff Lane Primary School we had sent in the following;

There are 28 triangles to be made. My year 6 class started by drawing the triangles from one peg. There were 7 triangles possible connecting peg 1 and 2, keeping that first short line constant. They then kept peg 1 the same and connected it to peg 3, keeping this constant. They were able to make triangles using pegs 4,5,6,7,8 and 9. This results in 6 possible triangles. They repeated for a constant line from peg 1 to peg 4 and found 5 triangles. They recognised the following pattern:

Constant line
Peg 1 - 2           7 triangles
Peg 1 - 3.          6 triangles
Peg 1 - 4.          5 triangles
Peg 1 - 5.          4 triangles
Peg 1 - 6.          3 triangles
Peg 1 - 7.          2 triangles
Peg 1 - 8.          1 triangle
Peg 1 - 9.          0 triangles

They added these up and found 28 triangles. This could be repeated for each new peg, but the triangles would be repeated, so would not count in the final total.

Answer 28

These are well thought-out answers.  It looks as if Theodore and the pupils from Cliff Lane Primary had alternative meanings of "different triangles" compared with Cong. What do you think?