Untangle — Always Solvable, Rarely Easy
Drag points until no lines cross. Every puzzle is built from a crossing-free layout — and the obvious method clears 95% of the mess then stops dead.
Drag the points until no two lines cross. Crossed lines are shown in red. A solution always exists — the puzzle was built from one.
It cannot be impossible, and it always has 3n − 6 lines
Each puzzle starts as a triangle. A new point is dropped inside one of the triangular faces and joined to that face's three corners, over and over. Since the point lands inside a face, the drawing never picks up a crossing — so the untangled answer is not something to hope for, it is literally what the puzzle was made from before the points were flung round a circle.
That construction also pins the line count. Every new point brings exactly three lines, so a puzzle on n points always has 3n − 6 of them: 12 lines at 6 points, 24 at 10, 48 at 18. And that is the ceiling for any flat drawing. At four points 3n − 6 gives 6, which is every possible connection. At five it gives 9 — one short of the ten that would join all five to each other, which is exactly the graph famously impossible to draw without a crossing.
The obvious method clears 95% and then stops dead
Play it the way anyone would: scan the points, drag whichever one improves things most, repeat until nothing helps. Run that to exhaustion on random puzzles and it does the bulk of the work almost instantly — and then refuses to finish.
| Points | Crossings at the start | Left when it stalls | Share cleared | Solved outright |
|---|---|---|---|---|
| 6 | 9.2 | 1.1 | 88% | 87% |
| 8 | 26.5 | 1.8 | 93% | 33% |
| 10 | 56.7 | 3.1 | 95% | 14% |
| 12 | 96.9 | 5.2 | 95% | 6% |
| 14 | 147.0 | 6.8 | 95% | 6% |
| 18 | 286.0 | 11.2 | 96% | 0% |
The striking column is the fourth. The share cleared barely moves — about 95% whether that means removing 25 crossings or 275. What is left over grows, but only in step with everything else.
Those last few crossings are stuck because no single point can be moved anywhere that improves the count. Untangling them needs two or three points rearranged together, and one-at-a-time improvement cannot see that far. It is a local minimum in the plainest sense.
Which is the useful thing to know as a player: being nearly finished is not the same as being nearly there. Down to two or three stubborn crossings, another small nudge is almost never the answer — you have to be willing to make things briefly worse.
The other instinct does not rescue it either
The natural alternative is to stop thinking about crossings and just let every point drift to the middle of its neighbours, repeatedly, pinning the outermost few so the whole thing does not collapse to a dot. That is the “Spread out” button above, and it is genuinely good on small puzzles — it finishes almost every 6-point board outright, where step-by-step improvement manages 87%. But it fades just as fast as the puzzles grow, solving under a tenth of the 18-point ones. Neither instinct is reliable, which is what makes the game worth playing rather than automating.
How to use
- Drag any point to move it; crossed lines turn red.
- Keep going until nothing is red.
- Spread out lets every point drift towards its neighbours.
- Show the answer reveals the layout the puzzle was built from.
Frequently asked questions
Can an Untangle puzzle be impossible?
No, and not merely as a matter of theory. Each puzzle is built by dropping points inside triangles of an existing crossing-free drawing, so a clean layout is what it was made from before the points were scattered. The answer button shows you that original.
How many lines does an Untangle puzzle have?
Always 3n minus 6, where n is the number of points — 24 lines at 10 points, 48 at 18. Every point added during construction brings exactly three lines, and that total happens to be the most any flat drawing can carry.
Why is 3n minus 6 the maximum?
Because it is the point where adding a single further line anywhere would force a crossing. At four points it allows six lines, which is every possible connection. At five it allows nine — one short of the ten needed to join all five to each other, and that graph is the classic example of one that cannot be drawn flat.
Why do the last few crossings feel so much harder than the first fifty?
Because they are. Improving one point at a time clears about 95 per cent of the crossings and then stalls completely, at every puzzle size measured. What remains is stuck because no single point can be moved anywhere that helps — those crossings need two or three points rearranged together.
Does that mean small nudges stop working near the end?
Essentially yes, and it is the most useful thing to know while playing. Being nearly finished is not the same as being nearly there. With two or three stubborn crossings left, the answer is usually to rearrange a whole cluster and accept that things look worse for a moment.
What does the Spread out button do?
It repeatedly moves every point to the average position of its neighbours, pinning the outermost few so the drawing does not collapse to a dot. It is surprisingly good on small puzzles, finishing almost every six-point board, but it fades quickly and solves under a tenth of the eighteen-point ones.
Is there a reliable way to solve these automatically?
Neither of the two obvious approaches is reliable. Step-by-step improvement stalls at a local minimum, and neighbour-averaging works well only when the puzzle is small. That gap between a guaranteed solution and an easy one is what makes the game worth playing.
Does this send anything anywhere?
No. Puzzles are generated and checked entirely in your browser.
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