What is the planarity game?
The planarity game starts with a tangle: points joined by lines, crossing each other all over the board. You can drag the points wherever you like, and the lines follow. The goal is to move the points so that no two lines cross. When the last crossing disappears, the knot becomes a neat, flat network.
It is the kind of puzzle you can play by feel, but it rewards a plan. Play the planarity game online free here, from a 6-point warm-up to a 25-point tangle.
Where Planarity comes from
Planarity was released in 2005 as a free Flash game by John Tantalo, based on a concept by Mary Radcliffe. It became popular quickly and led to many copies, often under the name untangle game or untangle puzzle. Since Flash no longer runs in browsers, versions like this one keep the puzzle playable.
How to play here
Pick the number of points on the start card, then press Play. Click or touch a point and drag it to a new spot. You can place points anywhere on the board, and you can drag the same point as often as you want.
Undo, Redo and New puzzle are in the dock. The pause menu has Restart, which puts the points back in their starting tangle, and Solve, which moves them into an untangled layout.
What a planar graph is
In maths, a set of points joined by lines is called a graph. A graph is planar if you can draw it on a flat sheet with no lines crossing. Some graphs can’t be drawn that way at all. A classic example is the three utilities puzzle: three houses each joined to three utilities, which always needs at least one crossing.
Every board in this game is planar, so a crossing-free drawing always exists. Better still, a result known as Fáry’s theorem says that any planar graph can be drawn with straight lines and no crossings. So you never need curved lines: some arrangement of the points will do it.
A step-by-step way to solve
- Clear some space. Drag a few points to the edges so the middle of the board is free to work in.
- Pick a starting shape. Find a triangle or a small loop and lay it out cleanly in the middle.
- Add points one at a time. Choose a point connected to your shape and put it where its lines don’t cross anything.
- Fix crossings as soon as they appear. One crossing is easy to fix, five at once is not.
Why play Planarity?
Planarity is relaxing, hands-on and easy to explain to anyone. Small boards take a minute; the bigger ones are a proper puzzle. For more puzzles about connections, try Pipe Game or Hashi.
How to play Planarity
- The board shows points joined by straight lines.
- You can drag any point anywhere. Its lines stretch and follow it.
- The connections never change. You only move points, you never add or remove lines.
- You win when no two lines cross anywhere on the board.
- Every puzzle can be solved, because each one is built from a graph that can be drawn without crossings.
Planarity tips
Find a triangle and build out
Look for three points that are all joined to each other. Spread them out, then place their neighbors one by one inside or around that triangle.
Park the easy points
Points with only one or two lines are the easiest to fix. Drag them close to their neighbors so they stop crossing everything else.
Give busy points room
A point with many lines is the hub of the drawing. Move it towards the middle of the free space and let its neighbors gather around it.
Frequently asked questions
How do you solve a planarity puzzle?
Pull a small group of connected points into a clean shape first, such as a triangle. Then add the remaining points one at a time, placing each one next to the points it connects to, and fix any new crossings as you go.
Is every planarity puzzle solvable?
Yes. Each puzzle is a planar graph, a network that can be drawn flat without any crossing lines. A theorem proved in the 1940s shows that such a graph can always be drawn with straight lines too.
Who made the original Planarity game?
Planarity was a Flash game written by John Tantalo in 2005, based on an idea by Mary Radcliffe. The same puzzle is often called Untangle.
