The Dividing Necklace

Exactly 4 distinct ways to cut the necklace: cuts at {0,4,8}, {1,5,9}, {2,6,10}, or {3,7,11}. Each produces 3 arcs of 4 beads, and each arc contains at least one bead of each colour. The 4 configurations are all distinct since the beads are fixed in order.

Label positions 0–11. To get 3 arcs of exactly 4 beads, cuts must be placed at positions that are exactly 4 apart: {a, a+4 mod 12, a+8 mod 12} for some starting position a. There are 12 choices for a, but many give the same physical cut. Step 1: The three cut positions must divide 12 into equal arcs of 4, so they are spaced 4 positions apart: {a, a+4 mod 12, a+8 mod 12}. Step 2: Distinct values of a that give different cut sets: {0,4,8}, {1,5,9}, {2,6,10}, {3,7,11}. Any a from 0-11 maps to one of these 4 sets (since a mod 4 determines the set). Step 3: Verify each cut gives balanced colours: Cut {0,4,8} gives arcs [0-3]=(R,B,G,R), [4-7]=(B,G,R,B), [8-11]=(G,R,B,G). Each arc has colours R,B,G plus one repeat — not exactly one of each, but at least one of each. Step 4: All 4 cut positions are valid (each arc has at least one R, B, G). Are they distinct as necklace divisions? Yes — rotating the necklace maps one cut to another, but since the necklace pattern has period 3 (not 4), different cut positions give visually different arc patterns. Step 5: Count: there are exactly 4 distinct ways to cut the necklace into 3 balanced arcs of 4 beads. Step 6: Under full rotational symmetry of the necklace (12 rotations), some cuts may become equivalent. But since the beads are fixed in place (the problem says 'fixed order'), all 4 are genuinely distinct. Key Insight: Necklace division problems combine modular arithmetic (spacing of cuts) with symmetry analysis (when are two cuts equivalent?). The fixed bead order in this problem removes most symmetry, making most cut positions genuinely distinct. If beads were identical within each colour, fewer distinct cuts would exist.