One Stroke

Edges once, nodes as often as needed

A one-stroke puzzle asks you to trace every displayed connecting line exactly once in one continuous route. Select a starting node, then move to a node joined to your current position by an unused edge. You may revisit nodes, but you may not reuse an edge or jump between unconnected nodes. The puzzle is complete when every edge has been traversed.

| Term | Meaning | |---|---| | Node (vertex) | A marked point where edges meet | | Edge | A line directly connecting two nodes | | Degree | The number of edges touching a node | | Odd node | A node with an odd degree | | Bridge | An edge whose early removal can separate unused parts of the graph | | Euler trail | A route using every edge exactly once |

The key distinction is between nodes and edges. Returning to the centre of a star several times is legal because each visit can use a different line. Travelling along the same line twice is not. Visual line crossings count only when a marked node connects them.

Undo removes the latest edge and returns you to the previous node. Restart clears the route. Both are useful, but a little analysis before choosing the first node prevents most failures.

Odd nodes, bridges, and route planning

Count odd-degree nodes first

At every node, count how many edges touch it. This number is the node’s degree. In any completed route, most visits pair one entering edge with one leaving edge. An even-degree node can pair all its edges. An odd-degree node has one unpaired edge, so it must serve as an endpoint.

That gives the central rule:

• With zero odd nodes, start anywhere; the route will finish where it began.
• With two odd nodes, start at one and finish at the other.
• With more than two odd nodes, no one-stroke solution exists under these rules.

Why can there not be exactly one odd node? Every edge contributes one degree to each of its two endpoints, so the total degree is even. Odd-degree nodes therefore always occur in even numbers.

Counting is most reliable when done systematically. Move clockwise around the diagram and count incident edges, not neighbouring dots. Two nodes may have several visual lines nearby but only the lines ending at that node contribute.

Do not cross a bridge too early

Choosing the correct start is necessary but not sufficient. A route can still strand unused edges. The usual cause is taking a bridge before its far-side region is ready to be completed.

Imagine two clusters joined by a single line. Once you cross that line, you cannot return across it. If unused edges remain in the first cluster, the route fails. Complete what you need on the current side before crossing, or enter the second cluster only when you can finish there.

A practical rule, known from Fleury’s algorithm, is: when several unused edges leave your current node, avoid an edge that would disconnect the remaining graph unless it is your only choice. You do not need formal graph analysis for every move. Ask a simpler question: “If I take this line now, can I still reach all the other unused lines from where I land?”

Use loops as self-contained units

Cycles are flexible because you can travel around a loop and return to the same junction. If a loop attaches to the rest of the drawing at one node, clear the whole loop when you are at that node, then continue through the larger route. Leaving one edge of the loop behind may force an impossible return.

At a busy central node, mentally group edges into excursions: a triangle that returns to centre, an outer square that returns, and a final path that exits toward the endpoint. Process closed excursions before taking the final exit.

This resembles the constructive Euler-route method: keep following unused edges until a loop closes, and insert any remaining loop when its shared node is reached. On small puzzles, thinking in loop-sized chunks is easier than memorising every individual move.

Reserve the finish

When there are two odd nodes, mark the destination before starting. Near the end, avoid entering that destination while it still has unused edges you cannot leave appropriately. Because it is supposed to absorb the route’s final unpaired edge, arriving there too soon can reverse the intended structure.

You can also work backward mentally from the finish. Which line is likely to be last? From which node must you approach it? Does that predecessor need other edges cleared first? This backward check often identifies a forced late sequence.

For an all-even puzzle, reserve a return edge into the starting node. If every route out of a region is consumed too early, you may finish at the wrong point despite having chosen a legal start.

Recalculate after each major section

Using an edge reduces the remaining degree of both endpoints by one. The unused graph can therefore have a different shape from the original drawing. After clearing a loop or crossing into a new region, pause and inspect only the untraced lines.

Completed edges are visual noise. Mentally erase them and count choices from your current node. If just one unused edge remains there, the next move is forced. If several remain, check which one preserves connectivity.

Diagnose a dead end precisely

If you arrive at a node with no unused exit while edges remain elsewhere, do not restart blindly. Undo to the most recent junction where you had a genuine choice. The route after that choice may have been forced, so changing an earlier arbitrary move gains nothing.

At that junction, identify why the chosen edge failed: it entered the final odd node early, crossed a bridge, or left part of a loop unfinished. Then select the alternative that directly fixes that cause. This turns trial and error into reusable understanding.

Mistakes that strand unused edges

Starting at the most central node automatically. Central nodes look convenient but may be even while two outer nodes are odd. Count degrees before selecting a start.

Thinking nodes cannot repeat. Repeated nodes are often essential. Track used lines; a return to a junction is legal through a different edge.

Treating every visual crossing as connected. A crossing without a marked node offers no turn. Follow the puzzle’s actual endpoints rather than the geometry alone.

Taking an attractive long edge first. It may be the only bridge between regions. Check whether unused lines would be stranded behind you.

Leaving half a loop for later. A loop attached at one junction is safest to clear as one excursion, returning to that junction before continuing.

Entering the intended finish too soon. In a two-odd-node puzzle, reserve the opposite odd node for the end unless unused edges allow a valid departure.

Restarting without locating the bad choice. Undo to the last branching decision and inspect the remaining graph. Most failed endings originate at a premature bridge or incomplete cycle.

Frequently asked questions

Can I visit the same node more than once?

Yes. Nodes may be revisited as often as the route requires. The restriction applies to edges: every connecting line must be used exactly once.

Where should I start a one-stroke puzzle?

Count each node's incident lines. If exactly two nodes have an odd number, start at either odd node and finish at the other. If every node is even, you may start anywhere and return to the start.

What if there are more than two odd nodes?

A connected graph with more than two odd-degree nodes cannot be drawn in one stroke without repeating an edge. A standard solvable level therefore has either zero or two odd nodes.

Do crossing lines create a node?

Only a marked connection counts as a node. Lines that merely cross visually do not connect unless the puzzle shows a selectable point at the crossing.