AI Search · Exam notes

Line Drawings: Junctions, Edge Labels, Consistency

The second CSP flavor asks 2D line drawings of trihedral objects: junctions and edges get labels from a fixed catalog, and the question is which junction labels 1 to 4 make every shared edge agree? One chain answers it: type each junction by arm count, rotate a catalog pattern onto it, and check each shared edge from both ends. No consistent labeling means NIL.

Points 1 and 2 are the rules. Point 3 works a four-junction drawing with two accepted answers. Point 4 has two drills to solve with a pen.

1. Catalog in one table

Edges take \(+\) (convex), \(-\) (concave), or an arrow (occluding edge; the head points along the edge). IN means the head points toward the junction, OUT away from it. In-plane rotations of every pattern are allowed. Read each pattern left arm first, then middle (if any), then right.

JunctionArms, left to right (middle = stem)
L1arrow OUT, arrow OUT
L2arrow OUT, minus
L3minus, arrow OUT
L4arrow IN, arrow OUT
L5arrow OUT, plus
L6plus, arrow OUT
W1arrow IN, plus, arrow IN
W2minus, plus, minus
W3plus, minus, plus
Y1arrow IN, arrow OUT, minus
Y2minus, minus, minus
Y3plus, plus, plus
T1arrow OUT, arrow OUT, arrow IN (stem)
T2arrow OUT, arrow OUT, arrow OUT (stem)
T3arrow OUT, arrow OUT, plus (stem)
T4arrow OUT, arrow OUT, minus (stem)

2. The three checks

  1. Type by arm count: 2 arms is L; 3 arms with an opposite pair is T (stem first); otherwise W or Y. Junctions 5, 6, 7 and free ends are never asked: their arms take whatever fits.
  2. Agree on shared edges: both ends must show the same base (\(+\), \(-\), arrow), and one arrow IN plus one arrow OUT, i.e. a single physical arrowhead. \(+\) meets only \(+\); \(-\) only \(-\) (these need no direction).
  3. Answer 1 to 4 or NIL: every edge between two asked junctions must agree. No full labeling for 1 to 4 means NIL. Dangling arms constrain nothing, which is why one drawing can admit two answers.

3. Worked example: four junctions, two accepted answers

3a. The drawing, redrawn (solve from this, then check the walkthrough)

1 2 3 4 5

Q75. Junctions 1, 2: L. Junction 3: W. Junction 4: T. Shared edges: 1-2, 2-3, 3-4.

Shared edgeJunction on each side
1-2Junction 1 (L) and junction 2 (L)
2-3Junction 2 (L) and junction 3 (W)
3-4Junction 3 (W) and junction 4 (T, stem up to 3)

Cover the walkthrough. Agree each shared edge from both ends; arms to junction 5 and free ends take what is left.

3b. Walkthrough: one shared edge per step, both official answers

Step 0 | filled chip = label placed on that arm, struck line = edge agrees both ways

Why this step

Tracker

3c. The question

Assign consistent labels to all the edges and junctions in the 2D line drawing above. Enter the labels of the junctions 1, 2, 3, 4 in order, or NIL if no consistent labeling exists.

3d. Answers first, then the working

Accepted answers: L4,L2,W2,T3 and L6,L6,W1,T3.

  1. Edge 1-2, first answer: junction 1 = L4 with IN on the 1-2 arm; junction 2 = L2 with OUT on the 1-2 arm. IN plus OUT is one physical arrowhead: agree.
  2. Edge 2-3, first answer: junction 2 = L2 with minus on the 2-3 arm; junction 3 = W2 with minus on the 2-3 arm. Minus meets minus: agree.
  3. Edge 3-4, both answers: W2 (or W1) puts plus on the 3-4 arm; junction 4 = T3 with plus on its stem. Plus meets plus: agree. Junction 4 is T3 in both answers.
  4. Edge 1-2, second answer: junction 1 = L6 with plus on the 1-2 arm; junction 2 = L6 with plus on the 1-2 arm. Plus meets plus: agree. Then W1 completes edges 2-3 (IN meets OUT) and 3-4 (plus meets plus).
  5. Why two answers: the 1-2 edge admits an arrow pair (IN + OUT) and a plus pair, and each choice propagates to a different consistent labeling. Arms to junction 5 and dangling ends take what is left over and constrain nothing asked.

4. Practice: two drills

Solve each with a pen before opening the answer. Each drill is self contained.

Drill J1: nested triangles (single key)

1 2 3

Junction 1: L (arms to 2, to 6). Junction 2: Y (arms to 5, to 1, to 3). Junction 3: L (arms to 2, to 4). Junction 4: W (arms to 5, to 3, to 6). Enter 1 to 4.

Step 0

Why this step

Tracker

Answer J1

L2,Y1,L3,W3. Edge 1-2: L2 minus meets Y1 stem minus. Edge 2-3: Y1 left IN-arrow meets L3 OUT-arrow as one physical arrowhead. Edge 3-4: L3 minus meets W3 middle minus. Remaining arms (to 5, to 6, free ends) take what fits: junction 4 puts plus on both.

Drill J2: one edge, two endings

Back on the Q75 drawing: junction 1 = L4 with IN on the 1-2 arm forces junction 2's 1-2 arm to carry what? And junction 3 = W2 with plus on the 3-4 arm forces junction 4's stem to carry what?

Step 0

Why this step

Tracker

Answer J2

OUT-arrow (one IN plus one OUT makes a single physical arrowhead; minus or IN would clash), so junction 2 is L2 with OUT on that arm. Plus (only plus meets plus; an arrow or minus on the stem kills it), so junction 4 is T3. Each shared edge is a two-way check: read the forced label, then keep only catalog patterns carrying it on the matching arm.

5. Exam checklist (write this on the rough sheet)

  1. Count arms first: 2 is L; opposite pair plus stem is T; otherwise W or Y.
  2. Only junctions 1 to 4 are graded. Make every edge between two asked junctions agree from both ends.
  3. Arrows: one IN plus one OUT per edge. Plus meets plus, minus meets minus, nothing else mixes.
  4. Minus/plus arms have no direction to check; arrow arms always do.
  5. Arms to unasked junctions and dangling ends take leftovers. Two full labelings means two accepted answers.
  6. Nothing fits anywhere means NIL. A single clashing edge is enough to kill a candidate.