{ "version": "1.0.0", "date": "2026-11-01", "creator": "ColorPage editorial team", "canonicalOwner": "https://coloringpagegenerator.net/four-color-theorem", "license": "CC BY 4.0", "scope": "Six original bounded polygon maps. Unbounded outside is not colored.", "maps": [ { "id": "corner-cross", "title": "Corners do not count", "regions": [ { "id": "A", "vertices": [ [ 0.0, 0.0 ], [ 120.0, 0.0 ], [ 120.0, 120.0 ], [ 0.0, 120.0 ] ], "label": [ 60.0, 60.0 ] }, { "id": "B", "vertices": [ [ 120.0, 0.0 ], [ 240.0, 0.0 ], [ 240.0, 120.0 ], [ 120.0, 120.0 ] ], "label": [ 180.0, 60.0 ] }, { "id": "C", "vertices": [ [ 0.0, 120.0 ], [ 120.0, 120.0 ], [ 120.0, 240.0 ], [ 0.0, 240.0 ] ], "label": [ 60.0, 180.0 ] }, { "id": "D", "vertices": [ [ 120.0, 120.0 ], [ 240.0, 120.0 ], [ 240.0, 240.0 ], [ 120.0, 240.0 ] ], "label": [ 180.0, 180.0 ] } ], "edges": [ [ "A", "B" ], [ "A", "C" ], [ "B", "D" ], [ "C", "D" ] ], "minimumColors": 2, "solution": { "A": 0, "B": 1, "C": 1, "D": 0 }, "checks": [ { "colors": 1, "colorable": false, "assignmentsChecked": 1 }, { "colors": 2, "colorable": true, "assignmentsChecked": 7 } ], "hint": "A and D meet at a point, so they may match. A and B share a side, so they must differ.", "whyMinimum": "At least two colors are needed because A shares a border with B. The checkerboard uses two." }, { "id": "three-way-meeting", "title": "Three neighbors", "regions": [ { "id": "A", "vertices": [ [ 0, 0 ], [ 240, 0 ], [ 120, 120 ] ], "label": [ 120.0, 40.0 ] }, { "id": "B", "vertices": [ [ 240, 0 ], [ 240, 240 ], [ 0, 240 ], [ 120, 120 ] ], "label": [ 150.0, 150.0 ] }, { "id": "C", "vertices": [ [ 0, 240 ], [ 0, 0 ], [ 120, 120 ] ], "label": [ 40.0, 120.0 ] } ], "edges": [ [ "A", "B" ], [ "A", "C" ], [ "B", "C" ] ], "minimumColors": 3, "solution": { "A": 0, "B": 1, "C": 2 }, "checks": [ { "colors": 1, "colorable": false, "assignmentsChecked": 1 }, { "colors": 2, "colorable": false, "assignmentsChecked": 8 }, { "colors": 3, "colorable": true, "assignmentsChecked": 6 } ], "hint": "Follow the three lines into the meeting point. Each pair shares a line, not just the point.", "whyMinimum": "A, B and C all share borders with each other. That triangle of neighbors needs three colors." }, { "id": "checkerboard", "title": "More regions, same two colors", "regions": [ { "id": "A", "vertices": [ [ 0.0, 0.0 ], [ 80.0, 0.0 ], [ 80.0, 80.0 ], [ 0.0, 80.0 ] ], "label": [ 40.0, 40.0 ] }, { "id": "B", "vertices": [ [ 80.0, 0.0 ], [ 160.0, 0.0 ], [ 160.0, 80.0 ], [ 80.0, 80.0 ] ], "label": [ 120.0, 40.0 ] }, { "id": "C", "vertices": [ [ 160.0, 0.0 ], [ 240.0, 0.0 ], [ 240.0, 80.0 ], [ 160.0, 80.0 ] ], "label": [ 200.0, 40.0 ] }, { "id": "D", "vertices": [ [ 0.0, 80.0 ], [ 80.0, 80.0 ], [ 80.0, 160.0 ], [ 0.0, 160.0 ] ], "label": [ 40.0, 120.0 ] }, { "id": "E", "vertices": [ [ 80.0, 80.0 ], [ 160.0, 80.0 ], [ 160.0, 160.0 ], [ 80.0, 160.0 ] ], "label": [ 120.0, 120.0 ] }, { "id": "F", "vertices": [ [ 160.0, 80.0 ], [ 240.0, 80.0 ], [ 240.0, 160.0 ], [ 160.0, 160.0 ] ], "label": [ 200.0, 120.0 ] }, { "id": "G", "vertices": [ [ 0.0, 160.0 ], [ 80.0, 160.0 ], [ 80.0, 240.0 ], [ 0.0, 240.0 ] ], "label": [ 40.0, 200.0 ] }, { "id": "H", "vertices": [ [ 80.0, 160.0 ], [ 160.0, 160.0 ], [ 160.0, 240.0 ], [ 80.0, 240.0 ] ], "label": [ 120.0, 200.0 ] }, { "id": "I", "vertices": [ [ 160.0, 160.0 ], [ 240.0, 160.0 ], [ 240.0, 240.0 ], [ 160.0, 240.0 ] ], "label": [ 200.0, 200.0 ] } ], "edges": [ [ "A", "B" ], [ "A", "D" ], [ "B", "C" ], [ "B", "E" ], [ "C", "F" ], [ "D", "E" ], [ "D", "G" ], [ "E", "F" ], [ "E", "H" ], [ "F", "I" ], [ "G", "H" ], [ "H", "I" ] ], "minimumColors": 2, "solution": { "A": 0, "B": 1, "C": 0, "D": 1, "E": 0, "F": 1, "G": 0, "H": 1, "I": 0 }, "checks": [ { "colors": 1, "colorable": false, "assignmentsChecked": 1 }, { "colors": 2, "colorable": true, "assignmentsChecked": 171 } ], "hint": "There are nine regions, but only horizontal and vertical neighbors matter. Try an alternating pattern.", "whyMinimum": "A and B force at least two colors. Alternating by row and column colors all nine regions with two." }, { "id": "even-ring", "title": "An even ring", "regions": [ { "id": "A", "vertices": [ [ 120.0, 76.32 ], [ 163.68, 120.0 ], [ 120.0, 163.68 ], [ 76.32, 120.0 ] ], "label": [ 120.0, 120.0 ] }, { "id": "B", "vertices": [ [ 120.0, 8.0 ], [ 232.0, 120.0 ], [ 163.68, 120.0 ], [ 120.0, 76.32 ] ], "label": [ 158.92, 81.08 ] }, { "id": "C", "vertices": [ [ 232.0, 120.0 ], [ 120.0, 232.0 ], [ 120.0, 163.68 ], [ 163.68, 120.0 ] ], "label": [ 158.92, 158.92 ] }, { "id": "D", "vertices": [ [ 120.0, 232.0 ], [ 8.0, 120.0 ], [ 76.32, 120.0 ], [ 120.0, 163.68 ] ], "label": [ 81.08, 158.92 ] }, { "id": "E", "vertices": [ [ 8.0, 120.0 ], [ 120.0, 8.0 ], [ 120.0, 76.32 ], [ 76.32, 120.0 ] ], "label": [ 81.08, 81.08 ] } ], "edges": [ [ "A", "B" ], [ "A", "C" ], [ "A", "D" ], [ "A", "E" ], [ "B", "C" ], [ "B", "E" ], [ "C", "D" ], [ "D", "E" ] ], "minimumColors": 3, "solution": { "A": 0, "B": 1, "C": 2, "D": 1, "E": 2 }, "checks": [ { "colors": 1, "colorable": false, "assignmentsChecked": 1 }, { "colors": 2, "colorable": false, "assignmentsChecked": 32 }, { "colors": 3, "colorable": true, "assignmentsChecked": 51 } ], "hint": "Alternate two colors around the four outside regions. What can you use for the center?", "whyMinimum": "A, B and C are pairwise neighbors, so two colors cannot work. The even outside cycle uses two colors; A uses a third." }, { "id": "four-mutual-neighbors", "title": "Four mutual neighbors", "regions": [ { "id": "A", "vertices": [ [ 120.0, 76.32 ], [ 157.828, 141.84 ], [ 82.172, 141.84 ] ], "label": [ 120.0, 120.0 ] }, { "id": "B", "vertices": [ [ 120.0, 8.0 ], [ 216.9948, 176.0 ], [ 157.828, 141.84 ], [ 120.0, 76.32 ] ], "label": [ 153.7057, 100.54 ] }, { "id": "C", "vertices": [ [ 216.9948, 176.0 ], [ 23.0052, 176.0 ], [ 82.172, 141.84 ], [ 157.828, 141.84 ] ], "label": [ 120.0, 158.92 ] }, { "id": "D", "vertices": [ [ 23.0052, 176.0 ], [ 120.0, 8.0 ], [ 120.0, 76.32 ], [ 82.172, 141.84 ] ], "label": [ 86.2943, 100.54 ] } ], "edges": [ [ "A", "B" ], [ "A", "C" ], [ "A", "D" ], [ "B", "C" ], [ "B", "D" ], [ "C", "D" ] ], "minimumColors": 4, "solution": { "A": 0, "B": 1, "C": 2, "D": 3 }, "checks": [ { "colors": 1, "colorable": false, "assignmentsChecked": 1 }, { "colors": 2, "colorable": false, "assignmentsChecked": 16 }, { "colors": 3, "colorable": false, "assignmentsChecked": 81 }, { "colors": 4, "colorable": true, "assignmentsChecked": 28 } ], "hint": "Every region shares a border with every other region. Try to find a pair that can reuse a color.", "whyMinimum": "All six pairs among A, B, C and D share a border. Four mutually adjacent regions require four different colors." }, { "id": "odd-ring", "title": "An odd ring", "regions": [ { "id": "A", "vertices": [ [ 120.0, 76.32 ], [ 161.5421, 106.5021 ], [ 145.6744, 155.3379 ], [ 94.3256, 155.3379 ], [ 78.4579, 106.5021 ] ], "label": [ 120.0, 120.0 ] }, { "id": "B", "vertices": [ [ 120.0, 8.0 ], [ 226.5183, 85.3901 ], [ 161.5421, 106.5021 ], [ 120.0, 76.32 ] ], "label": [ 157.0151, 69.053 ] }, { "id": "C", "vertices": [ [ 226.5183, 85.3901 ], [ 185.8319, 210.6099 ], [ 145.6744, 155.3379 ], [ 161.5421, 106.5021 ] ], "label": [ 179.8917, 139.46 ] }, { "id": "D", "vertices": [ [ 185.8319, 210.6099 ], [ 54.1681, 210.6099 ], [ 94.3256, 155.3379 ], [ 145.6744, 155.3379 ] ], "label": [ 120.0, 182.9739 ] }, { "id": "E", "vertices": [ [ 54.1681, 210.6099 ], [ 13.4817, 85.3901 ], [ 78.4579, 106.5021 ], [ 94.3256, 155.3379 ] ], "label": [ 60.1083, 139.46 ] }, { "id": "F", "vertices": [ [ 13.4817, 85.3901 ], [ 120.0, 8.0 ], [ 120.0, 76.32 ], [ 78.4579, 106.5021 ] ], "label": [ 82.9849, 69.053 ] } ], "edges": [ [ "A", "B" ], [ "A", "C" ], [ "A", "D" ], [ "A", "E" ], [ "A", "F" ], [ "B", "C" ], [ "B", "F" ], [ "C", "D" ], [ "D", "E" ], [ "E", "F" ] ], "minimumColors": 4, "solution": { "A": 0, "B": 1, "C": 2, "D": 1, "E": 2, "F": 3 }, "checks": [ { "colors": 1, "colorable": false, "assignmentsChecked": 1 }, { "colors": 2, "colorable": false, "assignmentsChecked": 64 }, { "colors": 3, "colorable": false, "assignmentsChecked": 729 }, { "colors": 4, "colorable": true, "assignmentsChecked": 412 } ], "hint": "Try alternating two colors around the five outside regions. Watch what happens at the last border.", "whyMinimum": "The outside five-cycle cannot alternate just two colors, so it needs three. A touches every outside region and needs a fourth." } ] }