Yes, the minimum color targets in Four Color Map Puzzle are independently verified for every map, and a winning assignment must use exactly that proven number of colors. Behind the scenes, a dedicated exhaustive oracle tests every possible coloring of each map with one, two, three, and four colors and records the smallest palette size that produces a valid map. That figure is the graph's chromatic number, which the puzzle displays as the disclosed target. Because the oracle inspects all 4 to the n assignments for a map with n regions, the target is not estimated, assumed from a single example solution, or copied from the Four Color Theorem. The puzzle's own test suite also confirms that the published literal solution is a proper coloring and uses exactly the chromatic number. Adjacency lists are validated for unique, symmetric, and self-loop-free entries before any coloring logic runs. This verification chain is what lets you treat the on-screen target as a hard constraint rather than a hint. When you color a region, you are racing against a fact, not a guess.
What the Minimum Color Target Actually Means
The minimum color target is the fewest number of colors required to properly color a given map's regions, which mathematicians call the chromatic number of the underlying adjacency graph. Four colors are an upper bound for any planar map by the Four Color Theorem, but most puzzles do not need all four. A simple two-region chain needs only two colors, a triangle needs three, and a fully connected four-region map forces all four. The minimum is not a suggestion. Using more colors than the chromatic number can still separate every adjacent pair, but the puzzle rejects that assignment because it misses the optimization goal. Each level in Four Color Map Puzzle therefore displays a single integer target, and you have to meet that exact count, not just keep neighbors different.
How the Exhaustive Oracle Verifies Each Target
Verification happens before gameplay begins. For each of the five maps, an independent exhaustive oracle enumerates every possible assignment using one color, then two, then three, and finally four. The first palette size that yields at least one proper coloring is recorded as the true chromatic number for that map, and the puzzle displays that number as the target. For a map with n regions, the oracle inspects up to all 4 to the n candidate assignments, which is small enough to run entirely in the browser. The oracle is not derived from the visible literal solution, and the literal solution itself must also pass two tests: every adjacent pair must use different colors, and the total number of distinct colors must equal the disclosed minimum.
Adjacency data is structurally checked before any coloring logic runs. Every neighbor index in each region's list must identify a real region. A region cannot list itself as a neighbor. No region can appear twice in a single neighbor list. Adjacency must be symmetric, so if region A lists B, then B must list A. Because the user interface draws each visible line from this symmetric data, duplicate lines cannot hide a malformed rule. The published academic material on planar coloring from Georgia Tech's Four Color Theorem page and the Carnegie Mellon conflict-graph notes describe the same region-to-vertex and border-to-edge model that these checks enforce. Those sources support the general mathematics only; the five maps, layouts, names, solutions, and score are original product fixtures.
The Five Maps and Their Proven Chromatic Numbers
The five fixed maps deliberately progress through different graph structures, so you encounter targets of two, three, three, four, and four colors across a single run. The table below shows each level's name, region count, proven target, and the structural reason behind that number.
| Level | Regions | Minimum Colors | Why |
|---|---|---|---|
| River Counties | 4 | 2 | Four-region chain; no odd cycle forces a third color. |
| Three Harbors | 3 | 3 | Triangle; every region touches both others, so two colors cannot alternate. |
| Island Ring | 5 | 3 | Odd five-cycle; an odd cycle cannot alternate on two colors. |
| Capital Districts | 4 | 4 | Planar drawing of the complete graph on four vertices; every pair is adjacent. |
| Sunwheel Provinces | 6 | 4 | Hub beside an odd five-region rim; the rim needs three, the hub forces a fourth. |
These targets are not guessed. They are the outputs of the exhaustive oracle described above, and each published literal solution uses exactly its minimum. The four-color upper bound from the Four Color Theorem remains true across the entire run, but it is only relevant for the last two levels, where the actual minimums happen to reach it. If you want to see the per-map palettes themselves laid out side by side, the Map Coloring Game: Five Maps With Proven Minimum Palettes guide walks through each level's coloring.
How to Play Four Color Map Puzzle
Each map is played the same way. Use a pointer or the arrow keys to move between regions, then pick one of four color choices and watch the score and mistake history react to the assignment.
- Select a labeled region with a click or the arrow keys. Left and Up move to the previous region, Right and Down move to the next.
- Choose one of the four numbered, named, and patterned color buttons. The number keys 1 through 4 apply the matching labeled color, and each palette choice shows a number, a name, and a symbol such as dots, stripes, crosses, or a grid mark.
- Color every region so that endpoints of every visible shared-boundary line use different colors, while using exactly the disclosed minimum number of colors for that map.
- Choose Check map or press Enter to submit the assignment. The score only changes when every region is filled and the assignment passes both the adjacency check and the exact-count check.
- Repair the first distinct invalid assignment by editing any region, then recheck. Repeatedly checking the same wrong assignment does not add another mistake to your history.
- After all five maps are verified, clear the run for exactly 1,000 points and have the best score saved locally in your browser.
Accessible Controls for Pointer, Keyboard, and Screen Reader
The same minimum targets are reachable without a mouse. Every region button is at least 44 pixels in both dimensions, exposes its region label, color name, and selected state to assistive technology, and the four color buttons expose the same data. Colored regions continue to show their letter and pattern, which means the game never relies on hue alone to tell you what you have already chosen. Keyboard controls cover the complete interaction: Left and Up move to the previous region, Right and Down move to the next, number keys 1 through 4 apply the 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, so a stray key never silently corrupts the run. The shared GameShell also provides a double-Escape boss key that replaces the puzzle with a spreadsheet-style cover and back again, useful when you want a quick visual reset.
Scoring, Mistakes, and the 1,000-Point Run
Each verified level awards exactly 200 points, and five levels therefore total exactly 1,000. A wrong assignment is recorded by its exact level and color sequence, and rechecking the same wrong assignment does not add another mistake. The first distinct wrong assignment remains editable so you can repair it; a second different wrong assignment on the same level deadlocks the run until you press R to restart. Restart returns you to River Counties with region A selected, every region uncolored, zero score, and no stored mistakes. Completion and deadlock both freeze selection, coloring, movement, reset, and further checks, which prevents accidental changes after the score is locked. The best completed score is stored locally when browser storage is available, so a perfect 1,000-point run stays visible across sessions without an account or upload. Every step above runs in the browser, so the exhaustive verification, the check-map contract, the local storage, and the boss key all work once the page has loaded.
For a deeper look, see Empty Shape in Fraction Shape Puzzle: Is It a Mistake?.