Solution
Balanced Necklaces
Show the problem again
A necklace has 8 beads equally spaced around a circle: exactly 4 black and 4 white. Two necklaces are the same if one is a rotation of the other (flipping over is not allowed). How many different necklaces are there? Give an integer.
Worked solution
The answer is 10. A rotation by $k$ positions splits the 8 places into $\gcd(k, 8)$ cycles each of length $d = 8/\gcd(k,8)$; a fixed colouring must be constant on cycles, so it needs $4/d$ black cycles out of $8/d$, possible only when $d \mid 4$. Group the rotations by $d$: $d = 1$ (identity, 1 rotation) fixes $\binom{8}{4} = 70$; $d = 2$ (1 rotation) fixes $\binom{4}{2} = 6$; $d = 4$ (2 rotations) fixes $\binom{2}{1} = 2$ each; $d = 8$ (4 rotations) fixes none. Burnside: $(70 + 6 + 4)/8 = 80/8 = 10$.
Source: Standard application of Burnside's lemma; no traceable original source. Statement written for AxiomIQ.