Solution
Three-Colour Bracelets
Show the problem again
A bracelet consists of 6 beads equally spaced on a circular loop, each bead red, green or blue (not every colour need be used). Two bracelets are the same if one can be rotated and/or flipped over to match the other. How many different bracelets are there? Give an integer.
Worked solution
The answer is 92. Use Burnside's lemma over the dihedral group of order 12. Rotation by $k$ steps fixes $3^{\gcd(k,6)}$ colourings: $729 + 3 + 9 + 27 + 9 + 3 = 780$. Of the six reflections, three pass through two opposite beads (cycle structure: 2 fixed beads + 2 swapped pairs, so $3^4 = 81$ fixed colourings each) and three pass between beads (3 swapped pairs, $3^3 = 27$ each). Reflections contribute $243 + 81 = 324$. The number of orbits is $(780 + 324)/12 = 1104/12 = 92$.
Source: Standard application of Burnside's lemma; no traceable original source. Statement written for AxiomIQ.