Q7 — Waltz Line Labeling: One NIL, One All-L1
Two four-junction drawings — the first wipes out to NIL, the second labels every junction L1.
The Question
Same junction dictionary as the other papers (L1–L6, W1–W3, Y1–Y3, T1–T4, rotations allowed). Two drawings to label.
| # | Drawing | Official answer |
|---|---|---|
| 7.1 | Four-junction drawing | NIL |
| 7.2 | Four-junction drawing | L1, L1, L1, L1 (junctions 1, 2, 3, 4 in order) |
The Topic in 60 Seconds
Each junction must realize one legal dictionary pattern; shared edges must carry the same label (and arrow direction) at both ends. Empty domain ⇒ NIL; full consistency ⇒ list the labels.
Deep-dive: Waltz Line Labeling.
The L entries that matter here
| Label | Geometry |
|---|---|
| L1 | both edges carry arrows pointing into the corner |
| L4 | both arrows point away |
| L2/L3 | one arrow + one concave () |
| L5/L6 | one arrow + one convex () |
7.1 — The first drawing → NIL
Work the constraint cascade:
- Classify the four junctions from the drawing.
- Force the most constrained junction first — its pattern pins two or three edge labels outright.
- Propagate: each pinned edge strikes every pattern at its far junction that disagrees about arrow direction or type.
- The cascade reaches a junction whose remaining patterns all demand an edge label already fixed the other way at the neighbour end — domain empties → inconsistent → NIL ✓
No assignment survives because an odd cycle of forced arrows/chains cannot close: some junction always ends up needing to be two things at once. One worked kill is enough for full marks — you do not need to enumerate all combinations.
7.2 — The second drawing → L1, L1, L1, L1
All four junctions are L-junctions, and the consistent assignment is L1 at each — every edge is occluding, with arrows circulating so that each corner receives two incoming arrows:
- Every junction is an L (two edges each); domains start at six patterns.
- Pick any junction: L1 pins both its edges as arrows directed into it.
- Propagate: a neighbouring L now has one edge fixed as an arrow arriving from outside; patterns demanding a different direction there die, and L1 fits again — pinning the next edge inward.
- The pattern closes around the drawing with no contradictions — every junction realizes L1 ⇒ answer L1, L1, L1, L1 ✓
Watch the assignment lock in:
Tricks & Tips
Answers: 7.1 NIL · 7.2 L1,L1,L1,L1