Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures. The curvatures form an orbit of a 'thin group,' a subgroup of an algebraic group having infinite index in its Zariski closure. The curvatures that appear must fall into one of six or eight residue classes modulo 24. The twenty-year old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture. This is joint work with Summer Haag, Clyde Kertzer, and James Rickards. Time permitting, I will discuss some new results, joint with Rickards, that extend these phenomena to certain settings in the study of continued fractions.