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How to Solve One-Stroke (Trace-Every-Line) Puzzles
The one-stroke puzzle is the oldest doodle in the book: draw a shape without lifting your pencil, and without going back over any line you have already drawn. It looks like something you either fluke or you don't - but there is a genuine, provable trick to it. Learn to read a figure the way a solver does and you can glance at almost any shape and tell whether it can be traced, where to begin, and which lines you dare not save for last. This guide covers all of it - the one rule that decides everything, and the habits that stop you stranding a line.
What a one-stroke puzzle actually is
Strip away the decoration and a one-stroke puzzle is just a set of dots joined by lines. Your job is to draw one continuous stroke that covers every line exactly once. You may pass through the same dot as often as you like, but each line gets lit only once - no lifting, no retracing. Mathematicians call a path like this a Eulerian trace, after Leonhard Euler, who worked out the rule while puzzling over the bridges of Königsberg in 1736.
The important part is that whether a figure can be traced is decided entirely by its shape. No cleverness rescues an untraceable figure, and no traceable one is secretly impossible. So the first skill is not drawing; it is reading - and it comes down to counting.
The one rule: count the odd junctions
Look at any dot and count how many lines meet there - that number is its degree. A dot where three lines meet is "odd"; where four meet, "even". Now count the odd dots in the whole figure. Euler's rule says a connected figure can be traced in one stroke only if it has exactly zero or exactly two odd dots - nothing else works.
Here is the intuition. Every time your stroke passes through a dot it uses one line to arrive and one to leave - lines in pairs - so a middle-of-route dot always burns an even number of them. The only dots allowed to be odd are the two ends: the one you start from and the one you finish on. That is why the magic numbers are zero and two, never one, three, or four.
You don't have to count by eye. In Mochi Loop, the dots that gently pulse mark your valid starting points - the game has done the counting for you. Two pulsing dots means an open path, and those two are its odd junctions; a figure where every dot pulses is a closed loop with no odd junctions, so any dot can lead. Reading the pulse is reading the rule.
Where to start: on an odd junction, always
The count tells you the shape can be traced. It also tells you where to begin - the single most common thing beginners get wrong.
If the figure has two odd dots, you have no choice: start on one of them and you will finish on the other. Start anywhere else and you are guaranteed to strand a line - not bad luck, arithmetic. If the figure has zero odd dots (every dot is even), it forms a closed loop, so you can start anywhere and will always end up back where you began.
So the routine is simple: before you touch anything, find the odd dots. Two of them? Pick one and commit. All even? Free choice. If you have ever traced a shape and kept ending with one line dangling, this is almost certainly why.
Never save a line for last, never strand a branch
Knowing the start is half the battle. The other half is choosing the order of your lines so you don't cut yourself off. The golden habit here has a formal name - the heart of Fleury's algorithm - but in plain English: don't take a line that splits the remaining figure in two unless it is the only line you have left.
Picture two blobs of lines joined by a single connecting line - a "bridge". Cross it early and everything on the far side is marooned; you can never get back to the blob you left without retracing. So clear the blob you are in first, and cross the bridge only when nothing else on your side is left to draw. Bridges are crossed last, never on the way out.
The same logic covers little dead-end spurs - a line hanging off a single dot that goes nowhere. Handle those early, instead of leaving them until the end when reaching them means doubling back over lit ground.
Because Mochi Loop builds every figure backwards from a real solution, a board is always fair - it has a valid one-stroke route by construction. So experiment freely: drag back a dot to peel off a wrong turn, or restart and open from the other pulsing dot, until the line closes.
A simple solving routine
Put the pieces together and you get a routine that cracks most figures on the first honest attempt:
- Count the odd junctions. Zero or two means it can be traced; anything else can't.
- Start on an odd junction - one of the two if there are two, anywhere if there are none.
- Clear the dead-end spurs early, while you can still reach them cheaply.
- Stay inside a region and finish it before you cross any line that would strand the rest.
- Cross bridges last. Only take the one connecting line once your current side is fully lit.
- If a line keeps dangling, don't force it - restart from the other odd junction and try the route the other way round.
Notice that guessing never appears. A one-stroke puzzle is decided by shape alone, so if a route keeps failing, the answer is never to draw faster - re-read the figure and change where or how you started.
Common mistakes
- Starting on an even dot. With two odd junctions on the board, an even start dooms the trace before you draw a line. Find the odd dots first.
- Crossing a bridge too early. The moment you cut the figure in two, half of it is unreachable. Empty your current side before connecting to the next.
- Leaving spurs for last. Those dead-end tails are cheap to clear early and expensive to reach at the end.
- Retracing to "fix" a route. Going back over a lit line breaks the one rule. Undo the wrong turn and choose a different line instead.
- Ignoring diagonals. On richer figures, lines run corner to corner as well as up, down, and across - easy to miss when your eyes scan only the straight rails.
Where to go next
The counting rule stays abstract until you feel it, so the fastest way to learn is to trace a few dozen figures. Start on the small grids, where two pulsing dots make the odd-junction rule obvious, then work up to the dense boards where the order of your lines matters. Within a few puzzles you stop drawing hopefully and start reading a shape at a glance - the same "what is already forced?" instinct that powers the path and logic puzzles below.