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Four-Colour Map Puzzles: Colour Every Region Without a Clash
A four-colour map puzzle asks a deceptively simple question: can you paint a whole map so that no two regions sharing a border ever wear the same colour, using only four paints? It sounds as if it should trap you eventually - yet a famous mathematical promise says it never can. Grasp that promise and a couple of habits about where to start and how to unwind a knot, and these puzzles turn into a steady, satisfying tidy-up.
The rule, and why four colours are always enough
Everything flows from one rule: no two regions that share a border may be the same colour. That is it. You are handed a map carved into regions - little contiguous blobs - and your job is to colour every one so that neighbours always differ. Regions far apart can happily share a colour; the rule only looks at pairs that actually touch.
The catch, and the fun, is that you only get four paints. You obviously cannot hand every region its own shade, so the game becomes an exercise in reusing the same four crayons cleverly as they weave around the map. In Mochi Color those four are fixed on every board - pink, gold, mint and blue - and a single tap cycles a region through them, with a fifth tap clearing it back to blank so you can change your mind freely.
Here is the part that feels almost like a trick. No matter how tangled the map or how awkwardly its regions interlock, four colours are always sufficient for any flat map drawn on a page. This is the four-colour theorem, one of the most famous results in mathematics, and it means a valid solution is guaranteed before you place a single colour.
That guarantee changes how you play. You are never hunting for whether an answer exists - it always does - only for the order that reveals it. So a dead end is not "this map might be impossible" but "I have painted myself into a corner and need to back up." Mochi Color leans on this, running a solver over every board and discarding any layout it cannot prove four-colourable, so each map arrives clean, blank and solvable.
Handy to know: because a solution is guaranteed, there is nothing to lose by experimenting. Mochi Color has no timer, no move counter and no fail state - when two neighbours clash, both pulse a red outline, and recolouring either one clears it. The glow is a helper, not a penalty.
Start from the most hemmed-in region
The biggest improvement you can make is choosing where to begin. Do not start in an open, sparse corner - start with the most crowded region on the board, the one sharing a border with the most neighbours. A region touching five or six others is the tightest constraint, so pinning its colour first removes the most uncertainty in one move.
It is the same logic solvers use everywhere: tackle the hardest constraint while you still have the most freedom to satisfy it. Leave the busiest region for last and you often arrive to find its neighbours have claimed all four colours between them, leaving nothing legal for the centre. Colour the hub early and its neighbours fall into place around it.
Try this in the game: on any board in Mochi Color, paint the busiest region first and fan outward. A clash only ever outlines the two regions that actually match, never the ones you have already settled, so you can watch that hub hold the map together as its neighbours slot in around it.
Corners do not count
This is the rule beginners most often get wrong, and fixing it instantly makes maps easier. Two regions only count as neighbours if they share an actual edge - a real length of border. Regions that merely touch at a single corner point, meeting diagonally like squares on a chessboard, are not neighbours and may share a colour.
Get this wrong and you burn a precious fourth colour dodging a clash that never existed, tightening the board for no reason. Before deciding two regions must differ, check that they truly run alongside each other rather than kissing at a point - more often than you would think, a knot loosens the moment you spot that a diagonal touch was free all along.
Think in chains, and recolour the neighbour
When two neighbours end up matching, the instinct is to stare at one of them and agonise over what it "should" be. Usually the faster fix is the opposite: leave that region and recolour the neighbour instead. With four paints and only two regions in any clash, there is almost always a spare colour one tap away.
What makes this click is thinking in chains rather than isolated choices. Colours ripple: fixing one border nudges the next, which nudges the one after. Follow the wave - recolour along a run of neighbours and watch the clashes clear in sequence. On busier maps, work outward from one corner so the regions you have already settled stay locked as fixed neighbours while you paint into open ground.
How to backtrack cleanly
Every so often you reach a region whose neighbours have genuinely used all four colours, so nothing legal fits. Do not treat this as failure - it just means an earlier choice was too greedy, and the guarantee promises a way through. The skill is unwinding it cleanly rather than poking at regions until it works.
Back up to the nearest busy hub feeding the stuck spot, change its colour, and let the ripple flow outward again. Reshuffling one well-chosen upstream region shifts all its neighbours at once and usually frees the jam in a single move. If a whole cluster is fighting itself, the cleanest reset is to start it over - in Mochi Color, pressing R or Backspace clears the entire map so you can recolour from blank.
A simple solving routine
Put the habits together and you have a reliable loop for almost any map:
- Scan for the hub. Find the region touching the most neighbours and colour it first.
- Work outward from that hub or a corner, giving each new region a colour that differs from its painted neighbours.
- Check edges, not corners. Only worry about regions that share a real border; ignore diagonal touches entirely.
- Fix clashes by recolouring the neighbour - a single tap on the adjacent region almost always clears it.
- If a region has no legal colour, back up to the nearest hub, change it, and let the ripple flow again. Wipe and restart a hopeless cluster rather than patching it piecemeal.
Notice there is no guessing in the list. Because a four-colouring is guaranteed, every dead end is just an earlier choice to revisit, never a sign the map cannot be solved.
Where to go next
The quickest way to make all of this second nature is simply to colour a few maps. Start on the small early boards until spotting the busiest hub feels automatic, then let the grids grow and the recolour-the-neighbour reflex carry you through the tangles. If you enjoy the "find what is already forced" feel, the related logic puzzles below scratch the same itch.