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# A Chain of Eight Polyhedra

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Age 7 to 11

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Many of you sent solutions to this problem, although none of you tried to find more than one way of making a chain. Wilson from Beecroft Primary School sent in a diagram which shows one chain very clearly. In fact Wilson noticed that this is a loop, not just a chain:

Rachael, Chloe and Emilia from Ardingly College Junior School found a different chain:

Hexagonal prism

Hexagonal based pyramid

Pentagonal based
pyramid

Pentagonal prism

Cuboid

Cube

Square based
pyramid

Triangular prism

Zara, also from Ardingly College Junior School discovered another chain:

Triangular prism - Square based pyramid - cube - cuboid - pentagonal
prism - hexagonal prism - hexagonal based pyramid - pentagonal based
pyramid.

The links between these eight shapes are that between the
triangular prism and square based pyramid are the triangle. Next the
link is the square, then the square again. Between the cuboid and
pentagonal prism is the rectangle. Next the link is rectangle again then
the hexagon. Last of all the link is the triangle.

Do these make loops too?

Alex and Katie from Woodfall Junior
described how the links worked in their loop:

Cube connects to the cuboid by a square.

Cuboid connects to the
hexagonal prism by a rectangle.

Hexagonal prism connects to the
hexagonal based pyramid by a hexagon.

Hexagonal based pyramid
connects to the pentagonal based pyramid by an isoceles triangle.

Pentagonal based pyramid connects to the pentagonal prism by a pentagon.

Pentagonal prism connects to the triangular prism by a rectangle.

Triangular prism connects to the square based pyramid by an equilateral
triangle.

Square based pyramid connects to the cube by a square.

Then we asked you how many different ways there are to make a loop.

Danny, aged 11, explained how he went about it:

I made a table of what types of faces each shape has, and then worked out from that which shapes can go next to each other.
| Square face | Hexagon face | Triangle face | Pentagon face | Rectangle face |

Cube | Yes | No | No | No | No |

Hexagon-based pyramid | No | Yes | Yes | No | No |

Pentagonal prism | No | No | No | Yes | Yes |

Square-based pyramid | Yes | No | Yes | No | No |

Hexagonal prism | No | Yes | No | No | Yes |

Cuboid | Yes | No | No | No | Yes |

Pentagon-based pyramid | No | No | Yes | Yes | No |

Trianglar prism | No | No | Yes | No | Yes |

Shape | Goes with |

Cube | Square-based pyramid, Cuboid |

Hexagon-based pyramid | Hexagonal prism,Square-based pyramid,Pentagon-based pyramid,Trianglar prism |

Pentagonal prism | Pentagon-based pyramid, Hexagonal prism, Cuboid, Trianglar prism |

Square-based pyramid | Cube, Cuboid, Hexagon-based pyramid,Pentagon-based pyramid, Trianglar prism |

Hexagonal prism | Hexagon-based pyramid, Pentagonal prism, Cuboid, Trianglar prism |

Cuboid | Cube, Square-based pyramid,Pentagonal prism,Hexagonal prism,Trianglar prism |

Pentagon-based pyramid | Pentagonal prism, Hexagon-based pyramid,Square-based pyramid,Trianglar prism |

Trianglar prism | Hexagon-based pyramid ,Square-based pyramid,Pentagon-based pyramid,Pentagonal prism,Hexagonal prism,Cuboid |

Then I saw that a cube can only go between the cuboid and the square-based pyramid, because they are the only two other shapes with square faces, so I put those three together. Then I saw which shapes could go next to the square-based pyramid, which are the cube, cuboid, hexagon-based pyramid, pentagon-based pyramid and triangular prism. But I'd already counted the cube and the cuboid, so I made a decision tree with three branches, one for the hexagon-based pyramid, one for the pentagon-based pyramid and one for the triangular prism.

I made branches from those using all the shapes they could go with that I hadn't used before, and continued until either the branch couldn't go any further or I'd counted all 8 shapes. I made 27 branches using all 8 shapes.

Then I had to see if they could join back to the cuboid again. Only the prisms could, because the other shapes had either been used to start off the chain or didn't have any faces the same as the cuboid. So I had 17 branches with all 8 shapes that could join back to the cuboid to make a loop.

Fantastic, Danny! Thank you!