The Four Color Theorem states that no more than four colors are needed to color the regions of any planar map so that no two adjacent regions share a color. In the standard map-coloring convention, two regions count as adjacent only when they share a boundary segment of positive length, not when they merely touch at a single point. The Four Color Map Puzzle turns that statement into five compact, fully auditable levels. Each level is an original planar adjacency graph drawn as labeled regions connected by lines, and the player must color every region using exactly the proven minimum number of colors. An independent exhaustive oracle verifies that minimum for every map by trying every possible assignment for palette sizes one through four. The game runs entirely in the browser, requires no account, and provides text, symbol, pointer, and keyboard controls for every interaction.
The Four Color Theorem in Plain English
The theorem was first proposed in the nineteenth century and later proved with substantial computer assistance, as documented on Georgia Tech's Four Color Theorem page. In its everyday form it says that a flat map of contiguous regions can always be filled with four colors so that any two regions touching along a real border receive different colors. Carnegie Mellon's conflict-graphs course material shows how a region becomes a node and a border becomes an edge in the abstract graph used to reason about coloring. The game uses the same shared-segment rule, so point-only contact is deliberately drawn without a line and never counts as an adjacency.
Because the theorem is an upper bound, it tells you that four colors always suffice but does not command that every map use all four. A simple chain of four regions needs only two colors because it has no triangles. A triangle of three regions needs three because each one touches the other two. The game highlights that distinction by displaying the proven target for each level rather than always showing four.
How the Game Models Regions and Borders
Each level in the four color theorem game is stored as a list of labeled regions plus a symmetric adjacency list. The implementation checks every fixture before release: every neighbor index must identify a real region, a region cannot be adjacent to itself, a neighbor cannot be listed twice, and adjacency must be symmetric so that if region A lists B, then B lists A in return. The interface then draws one visible line for each confirmed adjacency, which means a duplicate entry in the data cannot create a stray line and a missing entry cannot hide a real border.
That line-based display is the player's contract. Two regions that share a drawn line are adjacent and must take different colors. Two regions that touch at a corner but show no line are not adjacent and may share a color. The five levels draw their graphs without edge crossings, including the central region of the four-region complete graph and the hub of the six-region wheel, so the planar layout used by the original theorem is preserved on screen.
Chromatic Numbers: Why Four Is the Ceiling, Not the Target
The fewest colors that can properly color a graph is called its chromatic number, and the four color theorem only says that number is at most four. The game treats the chromatic number as a per-level target. A winning assignment must use exactly that many distinct colors, and using extras is counted as a miss even when every neighbor pair is different.
Each of the five fixed levels has a different chromatic number, and the structure of the map dictates why. The table below summarizes the official minima, the number of regions, and the structural reason for the target.
| Level | Regions | Chromatic number | Why this minimum |
|---|---|---|---|
| River Counties | 4 | 2 | Four-region chain with no cycles, so it alternates |
| Three Harbors | 3 | 3 | Triangle where every region touches both others |
| Island Ring | 5 | 3 | Odd five-cycle that cannot alternate on two colors |
| Capital Districts | 4 | 4 | Planar drawing of the complete graph on four vertices |
| Sunwheel Provinces | 6 | 4 | Hub beside an odd five-region rim, so the hub forces a fourth |
Those numbers are not assumed from the displayed literal solutions. The independent exhaustive oracle for each map tries palette sizes one through four in increasing order and enumerates every possible assignment, up to all 4 to the n combinations for n regions. It accepts an assignment only when every adjacency joins different values, and the first palette size that yields any valid assignment is taken as the true minimum. Tests also require each published literal solution to be valid and to use exactly that minimum, so the level can be cleared in the game but never under-budget.
How to Play the Four Color Map Puzzle
Open the puzzle and pick the first level. Use the keyboard or the pointer for every action, because both paths cover the complete game.
- Select a region by clicking it or by pressing Left and Up to move backward and Right and Down to move forward through the labeled list. The selected region is highlighted and announced to assistive technology.
- Apply a color by clicking one of the four palette buttons or by pressing the number keys 1 through 4. Each palette entry shows a number, a name, a symbol (dots, stripes, crosses, or a grid mark), and the matching color, and the colored region continues to display its letter label and pattern.
- Color every region before judging the map. An incomplete assignment does not change the score or the mistake history, and Check map waits for full coverage.
- Keep every line's endpoints on different colors. Two regions joined by a visible line must take different colors, and the level's disclosed target tells you how many distinct colors to use overall.
- Press Enter or click Check map to judge the assignment. A correct, fully colored assignment using exactly the minimum palette awards 200 points. The first distinct wrong assignment stays editable, the second different wrong assignment deadlocks the run, and a deadlock or completion freezes selection, coloring, movement, reset, and further checks until Restart.
- Repeat for the remaining four levels, then Restart to clear all mistakes and return to River Counties with region A selected and the score at zero. The best completed score is stored locally when the browser allows it, and a full run of five correct levels always totals exactly 1,000 points because each level awards exactly 200.
For a level-by-level walkthrough of the literal solutions, see the Color the Map Puzzle guide, which complements the math focus used here.
Color Patterns, Symbols, and Keyboard Access
Every palette choice combines a number, a name, a pattern, and a hue so the four color theorem game does not depend on color vision alone. The four patterns are dots, stripes, crosses, and a grid mark, and the colored region keeps its letter label on top of the fill. Each region and color button is at least 44 pixels in both dimensions and exposes its region name, color name, and selected state to assistive technology. Invalid color numbers, invalid region indices, unknown movement commands, and any input after completion or deadlock are atomic no-ops, so a stray key never changes the run.
| Key | Action |
|---|---|
| Left, Up | Select the previous region |
| Right, Down | Select the next region |
| 1, 2, 3, 4 | Apply the matching labeled color |
| Enter | Check the current map |
| X | Reset the current map |
| R | Restart the full run |
Scoring, Mistakes, and the Boss Key
Each verified level awards exactly 200 points, so five correct levels always reach exactly 1,000. The Check map contract is strict: an incomplete assignment is ignored, a wrong assignment is recorded by its exact level and color sequence, and rechecking the unchanged assignment does not add another mistake. A second distinct wrong assignment deadlocks the run until Restart, which clears the score, the mistake history, and the map back to region A on River Counties. Completion and deadlock freeze selection, coloring, movement, reset, and further checks.
The shared game shell also provides a double-Escape boss key. Press Escape twice within the timing window and the puzzle is replaced by a spreadsheet-style cover, then twice again to return. The cover preserves the underlying state so a hidden session is not lost. All coloring, validation, exhaustive fixture testing, scoring, and storage run locally in the browser, so the four color theorem game works without an account, upload, paid call, or server-side puzzle evaluator once the page is loaded.