Q7 — Waltz Line Labeling: One Solvable, One NIL
A rare consistent drawing — junctions 1–4 label as L2, Y1, L3, W3 — while the second drawing wipes out to NIL.
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 | L2, Y1, L3, W3 (junctions 1, 2, 3, 4 in order) |
| 7.2 | Second line drawing | NIL |
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 — a fully consistent assignment means the answer is the label list itself.
Deep-dive: Waltz Line Labeling.
The dictionary entries that matter here
| Label | Geometry |
|---|---|
| L2 | one arrow pointing away from the corner + one concave () edge |
| L3 | one concave () edge + one arrow pointing into the corner |
| Y1 | arrows into the junction on two edges, on the third |
| W3 | on one diagonal, on the stem, on the other diagonal |
7.1 — A consistent labeling exists → L2, Y1, L3, W3
The official key assigns junctions 1, 2, 3, 4 = L2, Y1, L3, W3. How to verify it fast in the exam:
- Classify the four junctions from the drawing (here: two L's, one Y, one W).
- Start at the Y (junction 2) — Y1 pins arrows on two of its edges and on the third. That is the most constrained junction, so everything downstream follows from it.
- Propagate each pinned edge to its neighbour: an arrow leaving the Y arrives at an L corner, selecting the arrow+ pattern there (L2 if it points away from that corner, L3 if it points into it), and the edge feeds the W.
- Close at the W (junction 4): its diagonals take , its stem takes — exactly W3. Every junction holds one legal pattern and every shared edge agrees at both ends ⇒ consistent ⇒ the answer is the label list L2, Y1, L3, W3 ✓
Watch the assignment lock in:
7.2 — The second drawing → NIL
Same method: classify, force the most constrained junction, propagate. Here the forced labels reach a junction whose remaining patterns all demand an edge label already ruled out — the domain empties → NIL ✓
The kill typically lands like this:
- A T-junction's bar arrows pin two edges immediately.
- Propagation forces a neighbouring junction into a pattern that over-constrains their shared edge.
- The last junction needs, say, two arrows pointing into it (any Y pattern) but every remaining edge already carries its arrowhead (or a at the other end. No dictionary pattern survives → domain size zero → inconsistent.
Tricks & Tips
Answers: 7.1 L2,Y1,L3,W3 · 7.2 NIL