Skip to content
Lizely

Four Color Map Puzzle

Color five original planar maps with their independently proven minimum palettes using accessible text, symbol, pointer, and keyboard controls.

Privacy: your files never leave your device. All processing happens locally in your browser.

How to use

  1. 1.Select a labeled region with a click or the arrow keys, then choose one of the four numbered, named, and patterned color buttons.
  2. 2.Color every region so that endpoints of every visible shared-boundary line use different colors, while using exactly the disclosed minimum number of colors.
  3. 3.Choose Check map or press Enter; repair the first distinct invalid assignment, then clear all five maps for exactly 1,000 points.

About Four Color Map Puzzle

Four Color Map Puzzle turns the classic map-coloring question into five compact, fully auditable levels. Each level is an original planar adjacency graph presented as a map of labeled regions. A visible line means that two regions share a boundary segment. Assign colors so that both ends of every line are different. Every region must be filled before a completed assignment can be judged.

The Four Color Theorem states that no more than four colors are needed to color the regions of any planar map while keeping adjacent regions different. In the standard map convention, regions are adjacent when they share a real boundary segment. Merely touching at a single point does not count. The interface makes that relationship explicit: each region is a labeled node, and each shared boundary is shown as a line between two nodes.

Four colors are an upper bound, not a command to use all four on every map. A simple chain needs only two colors, while a triangle needs three. The mathematical quantity for the fewest colors that can properly color a graph is its chromatic number. Each level displays its proven target, and a winning assignment must use exactly that many colors. Using extra colors can separate all neighbors but still misses the optimization goal.

The five fixed levels deliberately progress through different structures. River Counties is a four-region chain with chromatic number two. Three Harbors is a triangle, so every region touches both others and three colors are necessary. Island Ring is an odd five-cycle, which cannot alternate with only two colors and therefore needs three. Capital Districts is a planar drawing of the complete graph on four vertices and requires four distinct colors. Sunwheel Provinces places a hub beside an odd five-region rim; the rim needs three colors and the hub touches every rim region, forcing a fourth.

Those minimum targets are not assumed from the displayed literal solutions. An independent exhaustive oracle verifies them. For each map, it tries palette sizes one through four in increasing order and enumerates every possible assignment, up to all 4^n combinations for n regions. It accepts an assignment only when every adjacency joins different values. The first palette size with any valid assignment is the true minimum. Tests also require each published literal solution to be valid and to use exactly that minimum.

The adjacency fixtures receive structural checks before gameplay tests. Every neighbor index must identify a real region. A region cannot be adjacent to itself. A neighbor cannot be listed twice. Adjacency must be symmetric: if A lists B, B must list A. The UI derives each visible edge once from that symmetric data, so duplicate lines cannot hide a malformed rule. The chosen layouts draw the graphs without edge crossings, including the central region of the four-region complete graph and the hub of the six-region wheel.

To play with a pointer, click a region, then choose a color. Each palette choice includes a number, name, and symbol: dots, stripes, crosses, or a grid mark. Colored regions continue to show their letter and pattern. This redundant coding means the game does not rely on hue alone. Every region button is at least 44 pixels in both dimensions and exposes its region, color name, and selected state to assistive technology.

Keyboard controls cover the complete interaction. Left and Up select the previous region; Right and Down select the next. Number keys 1 through 4 apply the matching labeled color. Enter checks the assignment, X resets the current map, and R restarts the full run. Invalid color numbers, invalid region indices, unknown movement commands, and inputs after completion or deadlock are atomic no-ops.

Check map has a strict contract. An incomplete assignment does not change the score or mistake history. A fully colored assignment wins only if all adjacent regions differ and the number of distinct colors equals the stated minimum. A wrong assignment is recorded by its exact level and color sequence. Rechecking the unchanged assignment does not add another mistake. The first distinct wrong assignment remains editable; the second different wrong assignment deadlocks the run until Restart.

Each verified level awards exactly 200 points. Five levels therefore total exactly 1,000. Completion and deadlock freeze selection, coloring, movement, reset, and further checks. Restart returns to River Counties with region A selected, every region uncolored, zero score, and no stored mistakes. The best completed score is stored locally when browser storage is available.

The shared GameShell also provides a double-Escape boss key. Press Escape twice within the timing window to replace the puzzle with a spreadsheet-style cover, then twice again to return. All coloring, validation, exhaustive fixture testing, scoring, and storage run in the browser. The game has no account requirement, upload, paid call, new package, or server-side puzzle evaluator.

Georgia Tech's Four Color Theorem page is the primary academic source for the theorem and shared-boundary definition. Carnegie Mellon mathematics course material independently explains the region-to-vertex and border-to-edge graph model, proper coloring, point-only contact, fewest-color objective, and four-color upper bound. These sources support the general mathematics only. The five maps, layouts, names, solutions, score, checks, and interface are original Lizely product fixtures.

Methodology & sources

Represent each original map as labeled regions plus symmetric adjacency lists. Validate indices, uniqueness, absence of self-loops, and reverse adjacency. Independently enumerate every assignment for k=1 through 4 colors and take the first feasible k as the chromatic number; require minima 2,3,3,4,4 and require each literal solution to be proper and use exactly its minimum. Exercise production state actions for all five solutions, exact 200-point increments and 1,000 total, incomplete and invalid atomic no-ops, arrow selection, numeric coloring, reset, repeated-assignment deduplication, second-distinct-error deadlock, restart, and terminal freeze.

Frequently asked questions

What is the goal of Four Color Map Puzzle?
Give every region a color, keep every pair joined by a shared-boundary line different, and use exactly the independently proven minimum number of colors shown for that map.
Why does a point-only contact not make two regions adjacent?
The map-coloring convention treats regions as adjacent when they share a boundary segment, not when they merely touch at an isolated point. The puzzle shows each counted adjacency as a line.
Are the minimum color targets verified?
Yes. An independent exhaustive oracle tries every assignment for one, two, three, and four colors. The first feasible palette size is the true chromatic number for that fixed original graph.

Related tools

Mini Games guides

View all