December 2004
This month’s puzzles are all based on a common theme: the 24 triangles which can be formed by different patterns of shading using 4 colours. You will need a set of the triangles to solve the puzzles; you can download a picture, print it onto paper or card, and cut the triangles out.
All the puzzles have the same two basic rules:
- Triangles must be fitted together so that only sides of the same colour touch.
- All the sides around the edge of a shape must be of the same colour.
For example, the small hexagon shows a valid arrangement of 6 triangles.
Shapes
Rearrange the 24 triangles in the large hexagon so that the two rules are obeyed.
Extension
Arrange the 24 triangles in the large hexagon into 4 small hexagons, each made from 6 triangles, so that each arrangement obeys the two basic rules.
Explain the logic that you use to solve this problem.
(You will find it easier to solve these puzzles if you apply some logic rather than just trying things at random!)
Numbers
If 5 colours were used to shade the triangles instead of 4, how many possible triangles would there be?
Note that when 3 colours are used in a triangle (e.g. Red, Green, and Blue) there are two possible triangles because the colours can be arranged in two different ways: going around clockwise, either RGB or RBG.
With two colours, however (eg. Red and Green) there is only one triangle because RGR and RRG are really the same: you can rotate one to obtain the other.
Algebra
If n colours were used to shade the triangles instead of 4, how many possible triangles would there be? (This will be a formula which should work for any value of n.)
(It will help to do the ‘Numbers’ puzzle first!)
Miscellaneous
The diagrams show three puzzles. In each case, the aim is to find an arrangement of the 24 triangles which creates the shape and obeys the usual rules. Only two of these puzzles can be solved. Can you say which one cannot be solved, and explain why?
Extension
The diagrams show a duplication puzzle and a triplication puzzle. The aim of the duplication puzzle is to make, with the usual 24 triangles, two copies of the 12-triangle shape both of which obey the usual rules. The aim of the triplication puzzle is to make three copies of the 8-triangle shape all three of which obey the usual rules.
One of these puzzles can be solved and the other cannot. Which one is which, and why?