.

| | | | |

| | V | |

| | | |

| V *---* V |

| / \ |

V *---* *---* V

/ \ / \

*---* *---* *---*

/ \ / \ / \

* *---* *---* *

\ / \ / \ /

*---* *---* *---*

/ \ / \ / \

* *---* *---* *

\ / \ / \ /

*---* *---* *---*

/ \ / \ / \

* *---* *---* *

\ / \ / \ /

*---* *---* *---*

\ / \ /

*---* *---*

\ /

*---*

Your task is to draw lines through it along the five indicated vertical axes, using three colours (e.g. Blue, Green and Red). Then rotate it by 120 degrees and do the same along what are now the corresponding vertical axes, using another three colours (e.g. Cyan, Magenta and Yellow). Then rotate it by another 120 degrees and do it again, using yet another three colours (e.g. Amber, Fawn and Purple). Can you do it in such a way that no two of the 19 small hexagons has the same combination of colours crossing it, or alternatively show that that cannot be done?

If it can be done, then any solution can obviously have its colours permuted within each of the three sets of three colours - e.g. I could replace all Red lines by Green lines and all Green lines by Red lines and it would still be a solution. Also, any solution can be reflected or rotated and the sets of colours exchanged correspondingly - e.g. I could reflect a solution left-to-right, replace Cyan/Magenta/Yellow lines by Amber/Fawn/Purple lines respectively and vice versa to get another solution. Regard two solutions as essentially identical if you can transform one into the other by a sequence of such colour permutations, rotations and reflections. Can you find two solutions that are not essentially identical, or alternatively show that all solutions are essentially identical?

Note that you don't have to actually draw diagrams with coloured lines to show a solution - I've chosen the colours to all have different initial letters, so you can indicate lines just by typing the letters into the hexagon. E.g. the following diagram shows the state after drawing a full set of Blue, Green and Red lines, a Cyan line and a Purple line:

.

*---*

/ \

*---* G.. *---*

/ \ / \

*---* R.. *---* B.. *---*

/ \ / \ / \

* GC. *---* G.. *---* G.. *

\ / \ / \ /

*---* RC. *---* B.. *---*

/ \ / \ / \

* G.. *---* GC. *---* G.. *

\ / \ / \ /

*---* R.. *---* BC. *---*

/ \ / \ / \

* G.. *---* G.. *---* GCP *

\ / \ / \ /

*---* R.. *---* B.P *---*

\ / \ /

*---* G.P *---*

\ /

*---*

The objective is then basically to end up with a different combination of three letters in each hexagon.

Gengulphus