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Lizely

One Line Draw

Trace an original eight-node Euler puzzle, cover every edge exactly once, and verify the finite solution set with accessible touch and keyboard controls.

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How to use

  1. 1.Inspect the eight-node graph and use odd-degree parity to decide whether A or D should be your starting endpoint.
  2. 2.Tap connected nodes, or choose nodes with the arrow keys and Space, to trace nine continuous edge steps without repeating a line.
  3. 3.Use Undo or Clear to repair the route, then choose Check trail or press Enter to earn exactly 1,000 points.

About One Line Draw

One Line Draw is an edge-tracing logic game built around a precise challenge: draw one continuous stroke that uses every visible edge exactly once. The fixed board has eight labeled nodes and nine undirected edges. Tap or click a node to begin, then choose a connected node for each next step. A valid finish contains ten node visits because nine edge traversals connect them. You may revisit a node, but a winning route may not reuse an edge.

That distinction is central. This game is not Fill One Line, Numberlink, or another grid puzzle in which the objective is to cover cells. Empty space and cells do not matter here. Only the nine printed graph edges count. Nodes can appear more than once in a correct route when separate edges meet there. The status panel therefore reports both distinct edges covered and total steps, making a repeated line visible as a gap between those two numbers.

The rules come from graph theory. A trail is a walk that does not repeat an edge. An Euler trail is a trail that uses every edge in the graph exactly once. For a connected undirected graph with exactly two odd-degree vertices, an open Euler trail exists and its endpoints are those two odd vertices. OpenStax supplies the trail and Euler-trail definitions, while the University of New South Wales graph-algorithms course independently confirms the odd-degree condition.

The puzzle graph itself is original Lizely content. Its central spine is A-B-C-D. A triangular loop B-E-F-B is attached to B, and a second triangular loop C-G-H-C is attached to C. A and D each have degree one. B and C each have degree four, and E, F, G, and H each have degree two. Therefore A and D are the only odd-degree vertices. That parity is a useful hint without exposing a whole route.

A complete solution cannot begin at an arbitrary node. Because the graph has exactly two odd-degree vertices, every winning open trail starts at A and finishes at D, or starts at D and finishes at A. The bridge-like spine also constrains the order of the two triangular loops. When traveling from A toward D, the B loop must be completed before leaving B for C, and the C loop must be completed before leaving C for D.

Each triangle can be traced in either direction. From B, the upper loop can be B-E-F-B or B-F-E-B. From C, the lower loop can be C-G-H-C or C-H-G-C. Those two independent direction choices produce four A-to-D solutions. Reversing them produces four D-to-A solutions, for eight directed Euler trails in total. An edge-mask depth-first search starts at every node and independently enumerates exactly those eight routes.

One concrete winning route is A-B-E-F-B-C-G-H-C-D. It visits B and C more than once, which is legal because each return uses a different edge. Its reverse, D-C-H-G-C-B-F-E-B-A, is also valid. Other winning routes swap the orientation of either triangular loop. The game accepts the entire verified finite solution set instead of comparing your work with a single memorized string.

Drafting is forgiving. The interface allows you to traverse a visible edge more than once so that an incorrect nine-step attempt can be inspected. A repeated edge becomes thicker like any used edge, but the distinct-edge counter stops increasing. Use Undo step to remove the latest node or Clear trail to erase the current route while preserving the run. Clicking nodes that do not share an edge does not extend the route and does not count as a mistake.

Check trail has a strict two-strike contract. Checking before the route contains nine steps reports an incomplete attempt and does not change the mistake counter. A nine-step route that repeats an edge necessarily leaves another edge unused; the first such completed error records one mistake but remains editable. You can undo or clear it and try again. A second completed error deadlocks the run until Restart. Any verified Euler trail awards exactly 1,000 points.

Completion and deadlock are terminal. Once either state is reached, node selection, cursor movement, undo, clear, and checking no longer alter the game. Restart restores the same auditable graph with no route, zero mistakes, zero score, and node A selected for keyboard play. The puzzle does not use random generation, so the solution-count proof and every regression test remain reproducible.

Keyboard controls mirror pointer controls. Use the arrow keys to cycle through the eight labeled nodes, then press Space to choose the selected node. Press Delete or Backspace to undo the most recent step, C to clear the whole current trail, and Enter to check. Every node is a native button with a spoken label that identifies the current endpoint and keyboard selection. Touch targets are at least 44 pixels, and the graph stays inside a bounded panel on a 390-pixel viewport.

The shared game shell displays the score, stores the best score locally, provides Restart, and includes the double-Escape boss key. Press Escape twice within the shell timing window to show the spreadsheet-style cover, then twice again to return. All route validation and graph enumeration run in the browser. The game needs no account, uploads no data, requests no permission, calls no paid service, and adds no package dependency.

The cited references define the general mathematics only. They are not sources for this graph, its node positions, route count, scoring, interface, wording, or mistake policy. Those are product-authored and separately tested. One Line Draw is recreational logic practice; a 1,000-point finish proves only that this fixed nine-edge puzzle was traced correctly. It is not an intelligence test or a measure of educational, psychological, or medical ability.

Methodology & sources

Model the original fixture as eight indexed nodes and nine unique undirected edges: spine A-B-C-D, upper cycle B-E-F-B, and lower cycle C-G-H-C. Validate endpoint parity by computing every node degree and requiring A and D to be the only odd-degree nodes. Independently build adjacency entries carrying edge IDs, then run a depth-first search from every node with a nine-bit used-edge mask; accept only routes that consume all nine bits. Require exactly eight directed routes, endpoints A and D, ten node visits per route, and exact agreement with the production enumerator. Validate gameplay separately: reject non-adjacent extensions, permit repeated-edge drafts, ignore incomplete checks, preserve editability after the first full error, deadlock after the second, award exactly 1,000 for any enumerated Euler trail, and freeze all input at completion or deadlock.

Frequently asked questions

What counts as a valid One Line Draw solution?
A valid solution is one continuous route that traverses each of the graph's nine edges exactly once. Nodes may be revisited, but edges may not be repeated.
Why can a correct route visit the same node more than once?
Euler trails restrict repeated edges, not repeated vertices. Several different edges meet at B and C, so a correct route returns to those nodes through unused lines.
Does the fixed puzzle have a verified solution set?
Yes. An independent edge-mask DFS finds eight directed Euler trails: four from A to D and their four reversals from D to A.
Will checking an unfinished trail count as a mistake?
No. Fewer than nine steps is reported as incomplete without penalty. The first completed wrong route remains editable; the second deadlocks the run.

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