The figure
A square with both diagonals drawn (an X) and a triangular roof on top. It has 5 dots (two bottom corners, two top corners, the roof tip) and 8 lines.
| Dot | Lines meeting | Odd or even |
|---|---|---|
| Two bottom corners | 3 each (base, side, diagonal) | Odd |
| Two top corners | 4 each (top, side, diagonal, roof) | Even |
| Roof tip | 2 | Even |
Where to start
The only odd dots are the bottom corners, so you must start at one bottom corner and finish at the other (the odd-dot rule). Start at the roof tip or a top corner and some line is always left over, whatever order you try.
How many ways
There are 44 different drawing orders starting from the bottom-left corner. By symmetry there are 44 more from the bottom-right, 88 in total (we counted every order by computer).
One example: bottom-left → bottom-right → top-left (diagonal) → roof tip → top-right → top-left (top edge) → bottom-left (side) → top-right (diagonal) → bottom-right (side).
Other classic figures
| Figure | Lines | Odd dots | Start |
|---|---|---|---|
| Pentagram | 5 | 0 | Anywhere; you return to the start (2 ways from a given start, one per direction) |
| Bow tie (two triangles meeting at a point) | 6 | 0 | Anywhere |
| Envelope (the app’s version: a rectangle with a flap from the top corners to the middle) | 6 | 2 (top corners) | A top corner; 6 ways from each |
| X square (no roof) | 6 | 4 | Impossible; needs 2 strokes |
Envelopes are drawn differently by different people. Where the flap lines go changes which dots are odd, so count before you draw.
In the app
Classic figures in the Gallery
OneTrail’s Gallery has 26 classic figures (as of the launch version), each designed by hand and verified: the envelope, the House of Santa Claus, the bridges of Königsberg, the Star of David, a snowflake, a gear wheel and more. Each card shows its line count, and solved figures show your best time.
Turn off the starting-dot hint
Once you know the rule, switch off Show possible starting dots in settings and find the start yourself. A solve with no undos and no hints counts as a perfect solve.
FAQ
Can I find all 44 by hand?
Fix the start at the bottom-left and sort the orders by the first line you take (base, side or diagonal). That splits the search into three manageable groups.
What if I add a second roof?
Count again. The corners the new roof touches change from odd to even or back, so the starting point moves.
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.