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Course›Chapter 27›Lesson 27.4
Bonus chapter · puzzle studio 4

Pair Points with Every Required Segment Length

Use coordinates, right triangles, and exact matching to connect every marked point once—while building one segment of each required length.

Euclidean distancePythagorean triplesExact matchingCrossings allowedoriginal pages 158–159

Pair every point, and use every required length

Each segment uses two endpoints. Plan the whole matching so that no endpoint or required length is used twice. Crossings are allowed; sharing a stretch of a line is not.

  1. Learn on ten points: Missions 1–5 build and explain five segments of lengths 1–5.
  2. Try a new small board: Mission 6 uses eight points and lengths 1–4.
  3. Take on thirty points: Missions 7–8 use lengths 1–15. Three segments are already drawn, and you will add the other twelve.

The diagonal-distance tool prepares you for the larger board. On the two small matching boards, no diagonal has one of the required whole-number lengths.

The puzzle contract

One global plan, not fifteen separate guesses

A segment uses two points and one required length. Every choice removes those points—and may remove the only way to build a later length.

choose a rare length → reserve its endpoints → update the remaining choices

A locally correct segment can still make the whole puzzle impossible. Keep checking the remaining point–length system.

Distance toolkit — a new tool for diagonal segments
d² = (Δx)² + (Δy)²
3²+4²=5² · 6²+8²=10² · 5²+12²=13²

For a diagonal, draw a horizontal leg and a vertical leg. A right triangle obeys this rule: the square of its long side equals the sum of the squares of its legs. For legs 3 and 4, calculate 3×3 + 4×4 = 9 + 16 = 25. Since 5×5 = 25, the diagonal is 5 units long, not 3+4=7. Try the 3–4–5 preset below. Horizontal and vertical lengths are simply the change in the moving coordinate.

1
Read every condition

Separate crossing from overlapping

Not complete
●—●
Use every point onceEach marked point is an endpoint of exactly one segment.
╳
Crossing is allowedTwo non-collinear segments may pass through the same interior location.
═
Overlapping is forbiddenTwo collinear segments may not share a positive-length piece.
1…5
Use each target length onceThe worked example needs lengths 1, 2, 3, 4, and 5.
crossing: allowed overlap: forbidden
Important: the worked solution itself contains crossings. Rejecting every crossing would incorrectly reject the intended construction.
Rule check
2
Build a right triangle

Measure a segment from its coordinate changes

Not complete
Learn a tool for the larger challenge. A diagonal can have a whole-number length even though its endpoints are in different rows and columns. Use the horizontal and vertical changes to measure it; adding those changes would measure a bent route instead.
|Δx|12
|Δy|5
d²169
distance13
This is a 5–12–13 right triangle.
original-length check

The pre-drawn segment from (2,4) to (14,9) has length:

3
Worked example 3

Map the ten points and inspect candidate pairs

Not complete
Target length1
Candidate pairs2
Marked points10
Segments needed5
Required lengthNumber of candidate pairsStrategic meaning
Find the rarest length
4
Hands-on practice problem

Build all five segment lengths

Not complete
How to draw: choose one unused point, then choose another. The board accepts only an unused target length and rejects positive-length overlap. Crossing is allowed. One grid spacing is one unit. Diagonals are allowed if their actual length is a required whole number; this small board has no such diagonal pairs. A point counts as used only when it is an endpoint. Click a point, or focus it and press Enter or Space, then choose its partner.
Segments0 / 5
Points used0 / 10
Crossings0
Lengths remaining1, 2, 3, 4, 5
5
Turn a construction into a proof

Force the unique matching from rare lengths

Not complete
1 · Length 4
Only E–F has length 4, so that segment is forced.
2 · Length 1
The candidates are B–C and F–H. Since F is already used, B–C is forced.
3 · Length 3
The candidates are B–H and D–J. Since B is used, D–J is forced.
4 · Length 2
The only surviving length-2 pair is G–H.
5 · Length 5
The last two unused points are A and I, and AI=5.
Integer-length candidate edges11
Complete exact matchings1
First forced length4
Final pairA–I
Added audit: checking every perfect pairing of the ten original points confirms that the displayed construction is the only one using lengths 1 through 5 exactly once.
Uniqueness check
6
Added transfer puzzle

Apply the matching strategy on a new point set

Not complete
New challenge: pair eight points using lengths 1, 2, 3, and 4 exactly once. The board has one complete matching. One grid spacing is one unit. Diagonals are allowed in principle, but none of this board’s diagonal pairs has a required whole-number length.
Segments0 / 4
Points used0 / 8
Crossings0
Remaining1, 2, 3, 4
7
Read the larger puzzle

Prepare the thirty-point challenge

Not complete
Your larger puzzle: join 30 points into 15 segments, using each length from 1 through 15 exactly once. The segments of lengths 5, 10, and 13 are locked. They already use 6 endpoints, leaving 24 endpoints for your 12 new segments. The same crossing, overlap, and endpoint rules still apply.

This is the board you will solve in Mission 8. A complete matching is possible. First check how much remains to draw.

Optional investigation: can moving two points make the puzzle impossible?

Compare the main challenge with an altered layout under exactly the same rules. This tests whether having the right number of points is enough to guarantee a solution.

Points30
Locked lengths5, 10, 13
Complete matchings for this layout—
Selected layoutAltered
Moved point 1
(0,14) in the altered layout → (0,12) in the main challenge.
Moved point 2
(14,14) in the altered layout → (14,13) in the main challenge.
Run the search to test whether this layout can be completed.
Plan the remaining work
8
original Practice 3 studio

Complete all fifteen lengths on the main challenge board

Not complete
Your construction: lengths 5, 10, and 13 are already locked. Add one segment of each remaining length: 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, and 15. Use each of the 24 remaining endpoints once. A crossing uses no extra endpoint; overlapping segments are rejected.
Segments3 / 15
Points used6 / 30
Crossings0
Lengths remaining12 lengths
9
Independent practice

Matching and distance workshop

Not complete
1. What is the distance from (0,0) to (3,4)?
2. Under the worked example rules, may two non-collinear segments cross?
3. May two collinear segments share a positive-length section?
4. Which length is forced first in the ten-point original example?
5. How many complete exact matchings does the ten-point original example have?
6. Which three lengths are pre-drawn in Mission 8? Enter three whole numbers separated by commas, in any order.
8. A displacement of 12 units horizontally and 5 vertically makes what segment length?
Score: 0 / 8
10
Exit ticket

Show that you can plan a full point matching

Not complete
1. Each marked point should be an endpoint of…
2. Which expression gives the square of a segment’s length?
3. In the worked example, which pair makes length 2?
4. On the main challenge board, the length-8 segment runs from (0,4) to…
5. Which planning move is usually strongest?
Score: 0 / 5