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Chapter 26›Chapter 27›Lesson 27.5
Bonus chapter · Puzzle studio 5 of 9

Draw and Optimize Closed Orthogonal Paths

Connect dots with horizontal and vertical segments, close the path without crossing yourself, and turn every corner into a score. Then revise the whole construction—not just one corner—to push the total higher.

simple closed pathsright-angle products5 × 5 original board105-dot open challenge
Your responses, paths, and personal bests are stored in this browser.

Close a path, score it, then improve it

Your goal: draw a closed path made only of horizontal and vertical segments. It must not cross, touch, or overlap itself. At each turn, multiply the lengths of the two meeting segments; add these products for the score.

  1. Small 5×5 dot board, Missions 1–6: learn the rules and scoring from examples, then draw a legal closed path scoring at least 39 in Mission 5. Reaching 64 is a further challenge.
  2. Larger 105-dot cross, Missions 7–8: inspect a 341-point example, then draw your own legal closed path scoring at least 64. Matching or beating 341 is optional.

A benchmark is a score to compare with. A maximum requires proof that every legal path scores no higher.

The scoring rule

Multiply at each turn, then add

corner score = incoming segment length × outgoing segment length
total score = sum of all right-angle corner scores

Adjacent dots are one unit apart. Unused dots are allowed; diagonal segments, self-crossings, overlaps, and open paths are not.

Optimization mindset

Legal first. Better next. Proven best only with proof.

The worked example compares paths scoring 39 and 64, then presents a larger cross-board benchmark of 341 with 32 turns. It challenges the reader to beat 341; it does not prove that 341 is the maximum.

Keep the labels honest: “I found a path,” “this is my best path,” and “this is the greatest possible score” are three different claims.
1
Read the construction rules

Identify a legal closed orthogonal path

Not complete
↔ ↕
Orthogonal onlyEvery segment is horizontal or vertical.
▢
Close the loopThe final segment returns to the starting point.
∅
No self-contactNonadjacent parts may not cross, overlap, or touch.
···
Dots may remainA legal path need not use every point.
A · Open
B · Diagonal
C · Self-contact
D · Simple rectangle
Rule check
2
Score one turn correctly

Multiply the two segment lengths at a corner

Not complete
Incoming2
Outgoing4
Corner score8
Rulea × b
original connection: the corners marked D and E in Figure 27-9 each use lengths 2 and 4, so each contributes 2×4=8.
Corner and rectangle check
3
original Figure 27-9

Reconstruct the path that scores 39

Not complete
CornerA
Adjacent lengths3 × 2
Contribution6
Whole path39
The ten products are 6,2,2,8,8,4,2,2,2,3. The score belongs to the whole closed path, not to its area or perimeter alone.
original reconstruction
4
original Figure 27-10 + added exact audit

Stretch to the boundary and reach 64

Not complete
Read a worked rectangle. Five dots across make four unit gaps. Each side of the boundary square is therefore 4 units long. Calculate its score for the checkpoint; the search below is an optional investigation of whether any other legal path can do better.
4 corners × (4 × 4) = 64

The worked example’s second path uses the full width and height of the 5-by-5 point array. Each of its four turns contributes 16.

Optional: why 64 is the greatest possible small-board score

Try drawing in Mission 5 first if you want to explore. This search groups paths by the cells they enclose. Its counts describe all tested regions, not your current drawing.

Added proof tool: every legal simple orthogonal path on this board encloses a set of the 16 unit cells. The embedded audit checks all 2^{16}-1=65,535 nonempty cell selections, keeps those with one simple boundary, and scores their boundaries.
Cell selections checked—
Simple boundaries—
Greatest score—
Regions attaining it—
High scoreNumber of simple regions with that scoreInterpretation
Audit not run yet.
Read the rectangle
5
Hands-on 5 × 5 studio

Draw a legal path and improve its score

Not complete
Your small-board construction. Try to reach 39 points with a legal closed path. While the path is open, its first and last corners are unfinished, so the total is not scored yet.
How to draw: click a starting dot, then click an aligned dot. Consecutive segments must turn 90°. Click the starting dot again to close. The editor rejects diagonals, repeated vertices, overlap, crossing, and illegal self-touching. Complete this mission with a legal closed path scoring at least 39; then try to reach 64. Only turns count as vertices: skip dots along a straight segment. You may also focus a dot and press Enter or Space.
StateEmpty
Turns0
Closed-path score—
Personal best0

Close your path to calculate its score.

6
Optimization laboratory

See why full-span rectangles are powerful

Not complete
rectangle score = 4wh
One corner6
Four corners24
Area6
Board limit4 × 4
Do not use a false shortcut: adding more corners does not automatically add more score. A notch creates extra products but often shortens the segments beside them. Recalculate the complete total.
Optimize within the worked example board
7
original Practice 4 benchmark

Decode the 105-dot cross path that scores 341

Not complete
A new board, the same scoring rule. Study one completed path on this larger cross. Changing the selected turn changes the highlighted product; the full example always has 32 turns and scores 341.
Printed dots105
Turns32
Selected product2
Total341
2 × 1 = 2

How often each corner product occurs

Benchmark reading
8
Open optimization studio

Build, save, and compare paths on the 105-dot cross

Not complete
Stay on the cross: a segment may skip intermediate dots, but every unit gap it passes through must remain inside the plus-shaped dot array. This keeps each segment on one continuous row or column of the worked example board. Complete this mission with any legal closed path scoring at least 64. Matching or beating 341 is an optional further challenge.
StateEmpty
Turns0
Closed-path score—
Your score minus 341—

Close your path to calculate its score and compare it with 341.

Attempt history

9
Independent workshop

Audit your closed-path toolkit

Not complete
1. Which segment is permitted?
2. Lengths 4 and 7 meet at a turn. Its score is…
3. The score-39 path in Mission 3 has how many turns?
4. The full-boundary path in Mission 4 scores…
5. May some printed dots remain unused?
6. The practice cross contains… dots.
7. The worked example benchmark uses… turns.
8. Which statement is justified by the worked example?
Score: 0 / 8
10
Exit ticket

Explain what makes a path legal and high-scoring

Not complete
1. A 2-by-6 rectangle scores…
2. A path returns to its start but crosses itself. Is it legal?
3. At every scored vertex, the two incident segments meet at…
4. A full-boundary rectangle on the 5 × 5 point board scores…
5. Beating 341 on the cross would…
Score: 0 / 5