|Spoiler Inside: Solution to Puzzle||SelectShow>|
There are 22 pretty arrangements. Condition 3 is the unusual one.
The first and third groups of a three-group pattern must be the same colour and have four beads divided between the groups with at least one bead per group. One or more plus four equals three or less has no solutions. Condition 3 can never be met.
For symmetric patterns, there must be two beads of each colour in each half. There are six ways to arrange such beads in a half, so there are six symmetric patterns.
For the possible sequences of subpatterns, there are eight possibilities (showing length of the pairs’ groups) ([1 1 1 1], [1 1 2], [1 2 1], [1 3], [2 1 1], [2 2], [3 1], ). Double this because the subpatterns can be red then green or vice versa. This gives 16.
Conditions 1 and 2 are mutually exclusive. 6 + 16 = 22 pretty patterns