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How to Solve Untangle (Planar Graph) Puzzles
An untangle puzzle drops a knot of dots joined by strings in front of you, all crossing over one another, and asks one question: can you slide the dots around until nothing overlaps? It looks like chaos for a few seconds - and then you learn the mess is a promise. Every board can be laid flat, because it was built flat and then scrambled. This guide walks it back to that calm shape, from the one rule that defines a win to the moves that untie almost any tangle.
What an untangle puzzle actually is
Mathematicians call this a planarity puzzle. You are given a graph - dots (nodes) connected by lines (edges) - and your task is to arrange the dots so that no two lines cross. The connections never change; you only move the dots. When no line passes over another anywhere on the board, the graph is "planar" and you have solved it.
There is exactly one rule, and everything follows from it:
No two strings may cross. Move the dots until every crossing is gone.
The lovely part is that these puzzles do their own scorekeeping. You never count anything - a crossed string looks wrong and a clear one looks calm, so "solved" is simply the state where nothing looks wrong anymore.
See it live. In Mochi Untangle every string that crosses another burns bright red with a soft halo underneath, while clear ones fade to a muted line. The board becomes a heat map of your own mistakes: chase the red until the last one cools, and you have won - no counter, no timer, no way to lose a board.
Spread everything out first
The biggest mistake beginners make is trying to fix a tangle while it is still bunched up. When the dots sit in a tight clump, a dozen strings overlap in the same patch of screen and you cannot tell which line belongs to which dot. Do not solve from there.
Instead, your first job on any board is to fling the dots out wide. Grab each one and drag it toward the edge, giving every dot plenty of empty space. You are not solving yet - just untangling the visual mess so you can see the structure. A board that looked hopeless as a knot often reveals only two or three real crossings once it is spread out.
Aim for the outer ring
Here is the secret that makes these puzzles far less intimidating: the hidden solution is almost always a circle. Many untangle boards lay the dots evenly around a ring, add the connections so none cross in that arrangement, and only then scatter the dots. That means a tidy ring around the outside is a guaranteed answer sitting there in disguise, waiting to be rediscovered.
So once your dots are spread out, start nudging them toward an evenly-spaced circle around the edge, and leave the centre empty. You will not land the solved ring by luck, but working toward a ring does most of the heavy lifting - crossings melt away in bunches as the dots find the rim.
Feel the progress. Mochi Untangle plays a short chime that rises in pitch every time a drag removes a crossing, so a good move gets an audible pat before your eyes finish reading the board. Spread the dots wide, ease them toward the rim, and let the rising notes tell you you are on the right track.
Move one dot at a time and watch the colour
Once the board is open, resist the urge to shove three dots at once. Untangling is a game of small, deliberate moves. Pick the dot that touches the most crossed strings, slide it slowly, and watch which lines change as it goes. A single dot usually recolours a handful of strings, and the moment more of them settle than flare up, you have found its home - let go there.
If a drag ties as many new knots as it clears, put the dot back and try another direction. This one-dot-at-a-time discipline is what separates flailing from solving: you keep asking the same tiny question - "did that move leave fewer crossings than before?" - and follow the answer.
Work around the fixed anchors
Tougher boards introduce dots you cannot move. In Mochi Untangle these are the pinned mochi: they wear a white ring and a little peg, sit locked at their solved spot, and will not budge no matter how hard you drag. That sounds like a limitation, but it is really a gift - a pinned dot is a piece of the answer handed to you for free.
The trick is to read the fixed dots first, before you touch anything else. Note where each pegged dot sits, treat it as a corner of the finished shape, then arrange every draggable dot around those anchors rather than fighting them. Because a pinned dot is already in its final position, the strings running out of it point roughly toward where their other ends belong - follow those hints and the rest of the board falls into place.
A simple untangling routine
Put it together and you have a loop that unties the vast majority of boards:
- Spread out. Drag every movable dot away from the clump so the board has room and you can see individual strings.
- Read the fixed points. Spot any pinned or anchored dots and picture the finished shape around them.
- Push toward a ring. Ease the dots out to an evenly-spaced circle around the edge, leaving the middle clear.
- Attack the worst dot. Grab whichever dot still touches the most crossed strings and slide it until more lines settle than flare.
- Repeat and mop up. Keep picking the most-tangled dot and nudging it home until not a single string crosses another.
Notice there is no guessing and no counting in that list. You are simply making the board look tidier one dot at a time, and "tidy" and "solved" turn out to be the same thing.
Common mistakes to avoid
- Solving from the clump. Trying to fix crossings while the dots are still bunched is guesswork. Spread first, tidy second - always.
- Grabbing a random dot. Go after the dot with the most crossed strings, not the nearest one - the busiest dot clears the most knots per move.
- Moving too many dots at once. Big sweeping rearrangements make it impossible to tell what helped. One dot, one small move, then look.
- Fighting the anchors. Never plan around where you wish a pinned dot sat; read its real position first and build the layout to match.
- Ignoring the centre. If the middle stays crowded, strings there keep crossing. Push dots out to the rim and keep the centre open.
Where to go next
The quickest way to make this instinctive is to solve a few boards for real. Start on the early tangles where the spread-and-ring routine shines, then climb into the denser worlds and pinned dots that force you to plan around fixed points. Within a few puzzles you will stop seeing a scary knot and start seeing a circle in disguise. If you enjoy the "make it tidy and it is solved" feeling, the related guides below scratch the same itch.