Number points to draw a nonintersecting path
Give every circle a number. Join the numbers in order. Can your line visit every point without crossing or touching an earlier part of itself?
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A path may turn, but it may not cross itself
We use the center of each circle as a point. The grid helps you locate the points; it does not tell the line where to travel. Horizontal, vertical, and diagonal straight segments are all allowed if they pass every path rule.
Four rules, all at once
The double-ring circles in this lesson are fixed clues. You cannot erase, move, or exchange their numbers.
Use 1–10 on the small board, or 1–15 on the large board. Fill every circle, with one number in each.
Join 1 to 2, then 2 to 3, and so on. Do not join the last number back to 1. A path of 10 points has 9 segments.
Segments 1–2 and 2–3 may meet at point 2. Nonconsecutive segments may not cross, overlap, or touch. Consecutive segments may not overlap either.
A segment can cross grid lines and empty cells. But it cannot skip across a third point that also needs its own turn in the path.
A small but important note about the book’s wording
The worked example says that the line segments have “no common point.” Taken literally, that would forbid even the join at 2 between 1–2 and 2–3. The worked example drawings clearly include these joins. We therefore allow consecutive segments to share their intended numbered endpoint only.
The circles are markers, not solid obstacles: it is their centers that matter in the checker. This center-to-center convention is stated here rather than hidden in the code.
Look at the actual meeting point
Quick check
May segments 4–5 and 5–6 share point 5, with no overlap?
Follow a careful route through ten points
The worked example gives one successful numbering. Reveal it a little at a time, checking the new connections as they appear. These are useful choices—not claims that every choice is forced.
Rows run down; columns run across. Gold marks the newest connections.
Your path workshop
Tap a number, then a circle. Connections appear whenever both neighboring numbers are present. You can number in any order, but the path always follows numerical order.
Find a 10-point path
Every circle must be labeled. The last connection is 9–10.
Choose 3, then tap an empty circle. You can also use the coordinate control.
A candidate ring means the resulting currently drawn segments pass the checks. It does not promise that the whole puzzle can be finished. For an isolated number, no new segment appears yet, so a candidate ring gives no information about its future connections.
Select a number below, then tap a circle.
To move a placed number, choose it and tap a different circle. Two existing numbers swap. A new number replaces the old one. Fixed clues never move.
Keyboard help
Tab to a number and press Enter or Space to select it. Tab to a circle and press Enter or Space to place it. Arrow keys move focus among circles, in row order. Delete or Backspace clears a nonfixed circle.
The coordinate menu offers the same placements without aiming at the board. “Erase mode” stays on until you turn it off or choose a number.
Check these connections
Inspect a connection
Choose a connection to highlight it. A dashed placeholder in the list means one or both numbers are still missing.
Reveal the book’s 10-point path
Diagram 28–20. The checker does not require you to copy this numbering.
This replaces the current board. Undo brings your work back. Checkpoints record a checked path, not whether it was solved without help.
Try both original puzzles. Keep the printed numbers fixed. Fill the other circles, then connect consecutive numbers without intersections.
Example 8 · 1–10
Practice 7 · 1–15
Look beyond the very next line
Try three different positions for 3 on the small board. Each creates a legal first turn. But do all three choices leave a way to finish?
Where would you try 3?
The book does not analyze these branches. Here we test them with the same explicit path rules as the workshop.
When a choice fails, keep the rules and change the choice. Undoing a trial is the path-puzzle version of backtracking from the spacing-sequence lesson.
One answer—or the only answer?
A drawing proves that a solution exists. It does not prove that it is the only solution. And a familiar symmetry does not automatically preserve the fixed clues.
Find other valid numberings
Why this search covers every numbering
Fill labels in order. For a fixed label, try only its printed point. For any other label, try each unused point not reserved for a later fixed clue.
If a new segment crosses an earlier segment or passes through another point, reject that branch: adding later labels cannot repair an already drawn segment. Otherwise continue, then undo the trial and try the next point.
When every branch has been finished or ruled out, the list is exhaustive for the stated center-to-center model. These are calculations added for this lesson, not results asserted by the book.
What changes when we reverse the labels?
Use the small original path. Replace 1 by 10, 2 by 9, and so on. The geometric route stays the same, traveled the other way.
Reversal preserves a crossing-free route. It counts as a solution to the original puzzle only if every given label is still in the required circle.
Explain your thinking
Before you finish, check the ideas that make a path work—not just the final picture.
A good plan leaves room for every later point.
A finished answer passes every rule.
Four checkpoints complete
You have checked a valid path on both boards and explained the key rules. You can keep experimenting—your earlier checkpoints stay recorded.
Learning notes, exact conventions & teaching additions
In-lesson Practice 7: the same page, Figure 28–21.
original tasks: number the ten circles on a 5 × 5 board from 1 to 10, keeping 1, 2, and 10 fixed; then number the fifteen circles on a 6 × 6 board from 1 to 15, keeping 1, 2, 6, and 15 fixed. Connect consecutive numbers. The worked example supplies a diagrammed solution for each, but does not give a uniqueness proof or count all solutions.
Necessary interpretation: the statement’s “no common point” language cannot literally apply to the shared endpoint of consecutive segments. The worked example drawings show those intended joins. This page allows consecutive segments to meet only at their common numbered endpoint; other intersections, contacts, and overlaps are forbidden.
Explicit geometric model: circle centers are points. Every segment is straight and closed, including its endpoints. Passing through a third circle’s center is rejected even before that circle receives a label, because every center must eventually be visited. A marker’s decorative circumference is not an obstacle. Grid lines and empty cells are not obstacles. No segment joins the last point back to 1. Coordinates, double rings for given clues, arrows, and colors are added for clarity.
Checking and search: the workshop checks the given labels, all-number coverage, point uniqueness, third-point contacts, adjacent-segment overlaps, and intersections between nonconsecutive segments. A search stopped by its safety limit is reported as incomplete rather than as proof of impossibility.
Teaching additions: contact examples, the staged explanations of the worked example route, live checks, local-candidate highlighting, the three trial positions for label 3, contextual completion hints, exhaustive solution browsing, reversal testing, and the exit ticket.
Using the page: all styles, diagrams, rules, and calculations are included in this HTML file. No network requests are needed. Local browser storage can save boards, answers, and reflection; blocked storage does not prevent play. Work saves automatically when browser storage is available. The reset button removes this lesson’s saved work.