Recall that we have a number of coloured circles and wish to arrange them in such a way that any touching circles have different colours. Alice was able to create many arrangements that required just 3 colours, but Professor Pailyn challenged her to find a layout that had to have 4 colours.
Note that the golden circle is touching 3 circles with different colours and therefore it has to use a 4th colour. The minimum number of circles needed is thus 11. Try it yourself: set up the arrangement with coins and then try to colour them one by one using just 3 colours. You will reach a point where a 4th colour is necessary.
Drawing the above diagram was a little tricky until I figured out its geometric construction. I drew two regular hexagonal arrangements with six circles at the vertices and one in the centre. Then I rotated the two hexagons until two of the circles overlapped and the central pentagonal space was created. The resulting image has a vertical line of symmetry but the pentagonal space is not regular so there is no five-fold symmetry.
Hope you enjoyed this.
Original Question: Circle Patterns With Four Colours (PMQ40).
Many thanks for the inspiration from Professor Stewart's Hoard of Mathematical Treasures, by Professor Ian Stewart.
