Skip over navigation
Cambridge University Faculty of Mathematics NRich logo
menu search
  • Teachers expand_more
    • Early years
    • Primary
    • Secondary
    • Post-16
    • Events
    • Professional development
  • Students expand_more
    • Primary
    • Secondary
    • Post-16
  • Parents expand_more
    • Early Years
    • Primary
    • Secondary
    • Post-16
  • Problem-Solving Schools
  • About NRICH expand_more
    • About us
    • Impact stories
    • Support us
    • Our funders
    • Contact us
  • search

Or search by topic

Number and algebra

  • The Number System and Place Value
  • Calculations and Numerical Methods
  • Fractions, Decimals, Percentages, Ratio and Proportion
  • Properties of Numbers
  • Patterns, Sequences and Structure
  • Algebraic expressions, equations and formulae
  • Coordinates, Functions and Graphs

Geometry and measure

  • Angles, Polygons, and Geometrical Proof
  • 3D Geometry, Shape and Space
  • Measuring and calculating with units
  • Transformations and constructions
  • Pythagoras and Trigonometry
  • Vectors and Matrices

Probability and statistics

  • Handling, Processing and Representing Data
  • Probability

Working mathematically

  • Thinking mathematically
  • Developing positive attitudes
  • Cross-curricular contexts

Advanced mathematics

  • Decision Mathematics and Combinatorics
  • Advanced Probability and Statistics
  • Mechanics
  • Calculus

For younger learners

  • Early Years Foundation Stage

Nine-pin Triangles

Age 7 to 11
Challenge Level Yellow star
Primary curriculum
  • Problem
  • Getting Started
  • Student Solutions
  • Teachers' Resources

Cong who goes to St. Peter's RC Primary, Aberdeen, sent in a good 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

Richard, a teacher from Cliff Lane Primary School, sent in the following:

There are 28 triangles to be made.
My class started by drawing the triangles from one peg. There were 7 possible triangles 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. 

The next two solutions looked at a way of getting all the triangles by getting a series of numbers connected to the 9 points.
 

This is a good idea, and, as so often happens in mathematics as well as helping, can produce a problem. So while 123,124,125,126,127,128,129, as a starting point is a good idea the problem arises when you go to ones starting with 234,235,236,237,238,239 it looks like we have made a further six but the size and shape of 123 is identical to 234. The same is true for the ones that would follow on. It's a matter of not losing sight of what the numbers are representing.

The next solutions did well to see that 123,321,213,312,231,132 are all the same.



Here is Tiffany's, then Claire's, work from Citipointe Christian College in Australia:

   
 

 - - - - - - 
  

Jiwoo and Sushi then Harry, Judy and Mai from The British International School Ho Chi Minh City in Vietnam sent these in:







Thank you for these excellent thoughts and conclusions about this task, we hope to see you send in more solutions in the future.

Related Collections

  • Back to LTHC resources

You may also like

Tri.'s

How many triangles can you make on the 3 by 3 pegboard?

Cutting Corners

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

  • Tech help
  • Accessibility Statement
  • Sign up to our newsletter
  • Twitter X logo

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

University of Cambridge logo NRICH logo