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The single most useful trick for solving one-stroke puzzles — learn it once, use it forever
Before you start drawing, do this:
Step 1: Count connections at each point
Look at every intersection (vertex). Count how many lines touch it. Is it odd or even?
Step 2: Count the odd vertices
Now count how many vertices had an odd number of connections.
You'll end back where you started. This is an Euler circuit.
You must start at one of the two odd vertices. This is an Euler path.
No one-stroke solution exists. (In One Stroke, this never happens — every puzzle is guaranteed solvable.)
That's the entire rule. It sounds simple because it is. But it instantly tells you whether a puzzle is solvable and — critically — where to start drawing.
A --- B A: 2 edges (even)
| | B: 2 edges (even)
D --- C C: 2 edges (even)
D: 2 edges (even)Odd vertices: 0 → Euler circuit. Start anywhere, end where you started. Try: A→B→C→D→A ✓
A A: 2 edges (even) / \ B: 3 edges (ODD) ★ B---C C: 3 edges (ODD) ★ | | D: 2 edges (even) D---E E: 2 edges (even)
Odd vertices: 2 (B and C) → Euler path. Start at B, end at C (or vice versa). Try: B→A→C→B→D→E→C ✓
A D A: 2 edges (even) \ / \ B: 2 edges (even) \ / \ C: 4 edges (even) C F D: 2 edges (even) / \ / E: 2 edges (even) / \ / F: 2 edges (even) B E
Odd vertices: 0 → Euler circuit. Start anywhere. The center vertex C has degree 4 — it's the trickiest part, but the rule guarantees a solution exists.
A --- B A: 3 edges (ODD) ★ |\ /| B: 3 edges (ODD) ★ | X | C: 2 edges (even) |/ \| D: 4 edges (even) D---C (+ diagonal AC and BD)
Odd vertices: 2 (A and B) → Must start at A or B. This is why the envelope puzzle frustrates people who start at the wrong corner!
When you open a puzzle in One Stroke, here's the practical workflow:
Before drawing anything, quickly scan the grid for tiles with an odd number of connections. These are your mandatory start and end points. Starting anywhere else will lead to a dead end.
Once you know where to start, plan your path by working outward. Handle the "tricky" high-degree vertices (lots of connections) in the middle of your path, not at the beginning or end.
Even-degree vertices are "pass-through" points — you enter and leave them. If you enter an even vertex, you can always leave it. The danger is odd vertices, because if you enter one too early, you might get stranded there.
One Stroke's 3-tier hint system is designed for exactly this. A Tier 1 nudge often just reveals the starting direction — which aligns perfectly with what the odd/even rule tells you. Theory and hints reinforce each other.
Can I draw this in one stroke?
| Odd Vertices | Answer | Start Where? |
|---|---|---|
| 0 | Yes | Anywhere |
| 2 | Yes | At an odd vertex |
| 4+ | No | — |
Tip: In One Stroke, every puzzle has 0 or 2 odd vertices — guaranteed solvable.
Count the vertices with an odd number of connections. If there are 0, start anywhere (Euler circuit). If there are 2, start at one odd vertex (Euler path). If there are more than 2, the puzzle is impossible in one stroke.
Simply count every line (edge) that touches a point (vertex). If 3 lines meet at a point, its degree is 3 (odd). If 4 lines meet, its degree is 4 (even).
If there are 2 odd-degree vertices, start at one of them — you'll end at the other. If all vertices are even, start anywhere and you'll loop back to where you started. Starting at the wrong vertex is the #1 reason people get stuck.
When you pass through a vertex (enter and leave), you use 2 edges. So pass-through vertices need an even number of edges. Only your start and end vertices can be odd, because you leave the start without entering, and enter the end without leaving. This is Euler's theorem, proven in 1736.
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