The idea: work backwards
A figure drawn without lifting the pen carries its own answer: the order you drew it in. Erase the order and you have a puzzle that is guaranteed to be solvable. Count the dots of any such drawing and you will always find 0 or 2 odd ones.
Step by step
- Get dot paper: graph paper intersections, or dots 1–2 cm apart (a 4×4 to 5×5 grid).
- Draw in one go: in pencil, join dots with straight lines without lifting. Never use the same pair of dots twice.
- Mark start and finish: note them lightly for the answer sheet, then leave them off the puzzle.
- Copy it clean: redraw just the figure on fresh paper, or photocopy it.
- Check: count the lines at each dot and confirm there are 0 or 2 odd dots.
Tuning difficulty
| Lever | Easier | Harder |
|---|---|---|
| Number of lines | 8–12 | 30 or more |
| Finish | Back at the start (0 odd dots) | At a different dot (2 odd dots, start must be found) |
| Forks | Mostly dots with only 2 lines | Several dots with 4 or more lines |
| Dead-end traps | No small loops | Small loops that are hard to get back out of |
Forks and traps change how hard a puzzle feels far more than the line count does. The reason is explained in difficulty is not the number of lines.
Trick questions
Adding one line to a finished figure flips the odd/even status of both its ends. Add a line to a figure with 0 odd dots and you get 2 (still possible). Add a line between two even dots of a 2-odd-dot figure and you get 4 (impossible). Slip in one “this cannot be done” puzzle to check whether the rule has really sunk in.
How the app does it
Built and checked on your device
OneTrail does not ship a list of stored boards. It builds each board on the device and checks it can be solved before showing it. Journey boards use a fixed seed per world and stage, so stage 4-12 is the same board on every device.
Borrow ideas from Endless
Short of ideas for paper puzzles? In Endless, pick a board shape (Grid, Diagonal, Triangles, Radial, Mandala, Web), size, difficulty and whether to return to the start, and get a new board. Copy the ones you like onto dot paper.
FAQ
Can I use curved lines?
Yes, but a curve and a straight line between the same two dots count as two lines. For young children, straight lines are easier to count.
What about lines that cross?
Decide beforehand whether a crossing is a dot where you may turn, or just a place you pass over. If you mark it as a dot, its degree is 4.
OneTrail: One Line Draw Puzzle
No sign-up, fully offline, and every board is checked to be solvable before you see it. Free on Android.