diff --git a/package-lock.json b/package-lock.json
index 79ad20f..7614c2f 100644
--- a/package-lock.json
+++ b/package-lock.json
@@ -36,6 +36,8 @@
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         "@radix-ui/react-toggle-group": "^1.1.0",
         "@radix-ui/react-tooltip": "^1.1.4",
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+      "resolved": "https://registry.npmjs.org/@babel/runtime/-/runtime-7.28.4.tgz",
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@@ -156,6 +156,12 @@
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@@ -884,6 +890,24 @@
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+      "dependencies": {
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+        "@types/react-reconciler": "^0.26.7",
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+        "react-dom": ">=18 <19",
+        "react-native": ">=0.64",
+        "three": ">=0.133"
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@@ -2836,6 +3061,12 @@
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@@ -2922,11 +3159,16 @@
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@@ -2959,6 +3200,42 @@
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+      "resolved": "https://registry.npmjs.org/@types/react-reconciler/-/react-reconciler-0.26.7.tgz",
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+      "dependencies": {
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@@ -3201,6 +3496,12 @@
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+      "funding": [
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+          "url": "https://github.com/sponsors/feross"
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+          "url": "https://feross.org/support"
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+      }
+    },
+    "node_modules/stats-gl/node_modules/three": {
+      "version": "0.170.0",
+      "resolved": "https://registry.npmjs.org/three/-/three-0.170.0.tgz",
+      "integrity": "sha512-FQK+LEpYc0fBD+J8g6oSEyyNzjp+Q7Ks1C568WWaoMRLW+TkNNWmenWeGgJjV105Gd+p/2ql1ZcjYvNiPZBhuQ==",
+      "license": "MIT"
+    },
+    "node_modules/stats.js": {
+      "version": "0.17.0",
+      "resolved": "https://registry.npmjs.org/stats.js/-/stats.js-0.17.0.tgz",
+      "integrity": "sha512-hNKz8phvYLPEcRkeG1rsGmV5ChMjKDAWU7/OJJdDErPBNChQXxCo3WZurGpnWc6gZhAzEPFad1aVgyOANH1sMw==",
+      "license": "MIT"
+    },
     "node_modules/string-width": {
       "version": "5.1.2",
       "resolved": "https://registry.npmjs.org/string-width/-/string-width-5.1.2.tgz",
@@ -6855,6 +7448,15 @@
         "url": "https://github.com/sponsors/ljharb"
       }
     },
+    "node_modules/suspend-react": {
+      "version": "0.1.3",
+      "resolved": "https://registry.npmjs.org/suspend-react/-/suspend-react-0.1.3.tgz",
+      "integrity": "sha512-aqldKgX9aZqpoDp3e8/BZ8Dm7x1pJl+qI3ZKxDN0i/IQTWUwBx/ManmlVJ3wowqbno6c2bmiIfs+Um6LbsjJyQ==",
+      "license": "MIT",
+      "peerDependencies": {
+        "react": ">=17.0"
+      }
+    },
     "node_modules/tailwind-merge": {
       "version": "2.5.4",
       "resolved": "https://registry.npmjs.org/tailwind-merge/-/tailwind-merge-2.5.4.tgz",
@@ -6942,6 +7544,45 @@
         "node": ">=0.8"
       }
     },
+    "node_modules/three": {
+      "version": "0.160.1",
+      "resolved": "https://registry.npmjs.org/three/-/three-0.160.1.tgz",
+      "integrity": "sha512-Bgl2wPJypDOZ1stAxwfWAcJ0WQf7QzlptsxkjYiURPz+n5k4RBDLsq+6f9Y75TYxn6aHLcWz+JNmwTOXWrQTBQ==",
+      "license": "MIT"
+    },
+    "node_modules/three-mesh-bvh": {
+      "version": "0.7.8",
+      "resolved": "https://registry.npmjs.org/three-mesh-bvh/-/three-mesh-bvh-0.7.8.tgz",
+      "integrity": "sha512-BGEZTOIC14U0XIRw3tO4jY7IjP7n7v24nv9JXS1CyeVRWOCkcOMhRnmENUjuV39gktAw4Ofhr0OvIAiTspQrrw==",
+      "deprecated": "Deprecated due to three.js version incompatibility. Please use v0.8.0, instead.",
+      "license": "MIT",
+      "peerDependencies": {
+        "three": ">= 0.151.0"
+      }
+    },
+    "node_modules/three-stdlib": {
+      "version": "2.36.0",
+      "resolved": "https://registry.npmjs.org/three-stdlib/-/three-stdlib-2.36.0.tgz",
+      "integrity": "sha512-kv0Byb++AXztEGsULgMAs8U2jgUdz6HPpAB/wDJnLiLlaWQX2APHhiTJIN7rqW+Of0eRgcp7jn05U1BsCP3xBA==",
+      "license": "MIT",
+      "dependencies": {
+        "@types/draco3d": "^1.4.0",
+        "@types/offscreencanvas": "^2019.6.4",
+        "@types/webxr": "^0.5.2",
+        "draco3d": "^1.4.1",
+        "fflate": "^0.6.9",
+        "potpack": "^1.0.1"
+      },
+      "peerDependencies": {
+        "three": ">=0.128.0"
+      }
+    },
+    "node_modules/three-stdlib/node_modules/fflate": {
+      "version": "0.6.10",
+      "resolved": "https://registry.npmjs.org/fflate/-/fflate-0.6.10.tgz",
+      "integrity": "sha512-IQrh3lEPM93wVCEczc9SaAOvkmcoQn/G8Bo1e8ZPlY3X3bnAxWaBdvTdvM1hP62iZp0BXWDy4vTAy4fF0+Dlpg==",
+      "license": "MIT"
+    },
     "node_modules/tiny-invariant": {
       "version": "1.3.3",
       "resolved": "https://registry.npmjs.org/tiny-invariant/-/tiny-invariant-1.3.3.tgz",
@@ -6961,6 +7602,36 @@
         "node": ">=8.0"
       }
     },
+    "node_modules/troika-three-text": {
+      "version": "0.52.4",
+      "resolved": "https://registry.npmjs.org/troika-three-text/-/troika-three-text-0.52.4.tgz",
+      "integrity": "sha512-V50EwcYGruV5rUZ9F4aNsrytGdKcXKALjEtQXIOBfhVoZU9VAqZNIoGQ3TMiooVqFAbR1w15T+f+8gkzoFzawg==",
+      "license": "MIT",
+      "dependencies": {
+        "bidi-js": "^1.0.2",
+        "troika-three-utils": "^0.52.4",
+        "troika-worker-utils": "^0.52.0",
+        "webgl-sdf-generator": "1.1.1"
+      },
+      "peerDependencies": {
+        "three": ">=0.125.0"
+      }
+    },
+    "node_modules/troika-three-utils": {
+      "version": "0.52.4",
+      "resolved": "https://registry.npmjs.org/troika-three-utils/-/troika-three-utils-0.52.4.tgz",
+      "integrity": "sha512-NORAStSVa/BDiG52Mfudk4j1FG4jC4ILutB3foPnfGbOeIs9+G5vZLa0pnmnaftZUGm4UwSoqEpWdqvC7zms3A==",
+      "license": "MIT",
+      "peerDependencies": {
+        "three": ">=0.125.0"
+      }
+    },
+    "node_modules/troika-worker-utils": {
+      "version": "0.52.0",
+      "resolved": "https://registry.npmjs.org/troika-worker-utils/-/troika-worker-utils-0.52.0.tgz",
+      "integrity": "sha512-W1CpvTHykaPH5brv5VHLfQo9D1OYuo0cSBEUQFFT/nBUzM8iD6Lq2/tgG/f1OelbAS1WtaTPQzE5uM49egnngw==",
+      "license": "MIT"
+    },
     "node_modules/ts-api-utils": {
       "version": "1.3.0",
       "resolved": "https://registry.npmjs.org/ts-api-utils/-/ts-api-utils-1.3.0.tgz",
@@ -6987,6 +7658,43 @@
       "integrity": "sha512-jWVzBLplnCmoaTr13V9dYbiQ99wvZRd0vNWaDRg+aVYRcjDF3nDksxFDE/+fkXnKhpnUUkmx5pK/v8mCtLVqZA==",
       "license": "0BSD"
     },
+    "node_modules/tunnel-rat": {
+      "version": "0.1.2",
+      "resolved": "https://registry.npmjs.org/tunnel-rat/-/tunnel-rat-0.1.2.tgz",
+      "integrity": "sha512-lR5VHmkPhzdhrM092lI2nACsLO4QubF0/yoOhzX7c+wIpbN1GjHNzCc91QlpxBi+cnx8vVJ+Ur6vL5cEoQPFpQ==",
+      "license": "MIT",
+      "dependencies": {
+        "zustand": "^4.3.2"
+      }
+    },
+    "node_modules/tunnel-rat/node_modules/zustand": {
+      "version": "4.5.7",
+      "resolved": "https://registry.npmjs.org/zustand/-/zustand-4.5.7.tgz",
+      "integrity": "sha512-CHOUy7mu3lbD6o6LJLfllpjkzhHXSBlX8B9+qPddUsIfeF5S/UZ5q0kmCsnRqT1UHFQZchNFDDzMbQsuesHWlw==",
+      "license": "MIT",
+      "dependencies": {
+        "use-sync-external-store": "^1.2.2"
+      },
+      "engines": {
+        "node": ">=12.7.0"
+      },
+      "peerDependencies": {
+        "@types/react": ">=16.8",
+        "immer": ">=9.0.6",
+        "react": ">=16.8"
+      },
+      "peerDependenciesMeta": {
+        "@types/react": {
+          "optional": true
+        },
+        "immer": {
+          "optional": true
+        },
+        "react": {
+          "optional": true
+        }
+      }
+    },
     "node_modules/type-check": {
       "version": "0.4.0",
       "resolved": "https://registry.npmjs.org/type-check/-/type-check-0.4.0.tgz",
@@ -7128,6 +7836,15 @@
         }
       }
     },
+    "node_modules/use-sync-external-store": {
+      "version": "1.6.0",
+      "resolved": "https://registry.npmjs.org/use-sync-external-store/-/use-sync-external-store-1.6.0.tgz",
+      "integrity": "sha512-Pp6GSwGP/NrPIrxVFAIkOQeyw8lFenOHijQWkUTrDvrF4ALqylP2C/KCkeS9dpUM3KvYRQhna5vt7IL95+ZQ9w==",
+      "license": "MIT",
+      "peerDependencies": {
+        "react": "^16.8.0 || ^17.0.0 || ^18.0.0 || ^19.0.0"
+      }
+    },
     "node_modules/util-deprecate": {
       "version": "1.0.2",
       "resolved": "https://registry.npmjs.org/util-deprecate/-/util-deprecate-1.0.2.tgz",
@@ -7135,6 +7852,15 @@
       "dev": true,
       "license": "MIT"
     },
+    "node_modules/utility-types": {
+      "version": "3.11.0",
+      "resolved": "https://registry.npmjs.org/utility-types/-/utility-types-3.11.0.tgz",
+      "integrity": "sha512-6Z7Ma2aVEWisaL6TvBCy7P8rm2LQoPv6dJ7ecIaIixHcwfbJ0x7mWdbcwlIM5IGQxPZSFYeqRCqlOOeKoJYMkw==",
+      "license": "MIT",
+      "engines": {
+        "node": ">= 4"
+      }
+    },
     "node_modules/vaul": {
       "version": "0.9.9",
       "resolved": "https://registry.npmjs.org/vaul/-/vaul-0.9.9.tgz",
@@ -7230,11 +7956,21 @@
         }
       }
     },
+    "node_modules/webgl-constants": {
+      "version": "1.1.1",
+      "resolved": "https://registry.npmjs.org/webgl-constants/-/webgl-constants-1.1.1.tgz",
+      "integrity": "sha512-LkBXKjU5r9vAW7Gcu3T5u+5cvSvh5WwINdr0C+9jpzVB41cjQAP5ePArDtk/WHYdVj0GefCgM73BA7FlIiNtdg=="
+    },
+    "node_modules/webgl-sdf-generator": {
+      "version": "1.1.1",
+      "resolved": "https://registry.npmjs.org/webgl-sdf-generator/-/webgl-sdf-generator-1.1.1.tgz",
+      "integrity": "sha512-9Z0JcMTFxeE+b2x1LJTdnaT8rT8aEp7MVxkNwoycNmJWwPdzoXzMh0BjJSh/AEFP+KPYZUli814h8bJZFIZ2jA==",
+      "license": "MIT"
+    },
     "node_modules/which": {
       "version": "2.0.2",
       "resolved": "https://registry.npmjs.org/which/-/which-2.0.2.tgz",
       "integrity": "sha512-BLI3Tl1TW3Pvl70l3yq3Y64i+awpwXqsGBYWkkqMtnbXgrMD+yj7rhW0kuEDxzJaYXGjEW5ogapKNMEKNMjibA==",
-      "dev": true,
       "license": "ISC",
       "dependencies": {
         "isexe": "^2.0.0"
@@ -7525,6 +8261,23 @@
       "funding": {
         "url": "https://github.com/sponsors/colinhacks"
       }
+    },
+    "node_modules/zustand": {
+      "version": "3.7.2",
+      "resolved": "https://registry.npmjs.org/zustand/-/zustand-3.7.2.tgz",
+      "integrity": "sha512-PIJDIZKtokhof+9+60cpockVOq05sJzHCriyvaLBmEJixseQ1a5Kdov6fWZfWOu5SK9c+FhH1jU0tntLxRJYMA==",
+      "license": "MIT",
+      "engines": {
+        "node": ">=12.7.0"
+      },
+      "peerDependencies": {
+        "react": ">=16.8"
+      },
+      "peerDependenciesMeta": {
+        "react": {
+          "optional": true
+        }
+      }
     }
   }
 }
diff --git a/package.json b/package.json
index 47d0209..538fcb9 100644
--- a/package.json
+++ b/package.json
@@ -39,6 +39,8 @@
     "@radix-ui/react-toggle": "^1.1.0",
     "@radix-ui/react-toggle-group": "^1.1.0",
     "@radix-ui/react-tooltip": "^1.1.4",
+    "@react-three/drei": "^9.122.0",
+    "@react-three/fiber": "^8.18.0",
     "@tanstack/react-query": "^5.56.2",
     "@types/qrcode": "^1.5.5",
     "canvas-confetti": "^1.9.3",
@@ -61,6 +63,7 @@
     "sonner": "^1.5.0",
     "tailwind-merge": "^2.5.2",
     "tailwindcss-animate": "^1.0.7",
+    "three": "^0.160.1",
     "vaul": "^0.9.3",
     "zod": "^3.23.8"
   },
diff --git a/src/App.tsx b/src/App.tsx
index 7d2f5fb..d73976c 100644
--- a/src/App.tsx
+++ b/src/App.tsx
@@ -58,6 +58,7 @@ import ParityBitsGamePage from './pages/ParityBitsGamePage';
 import CrapsGamePage from './pages/CrapsGamePage';
 import NeighborSumAvoidancePage from './pages/NeighborSumAvoidancePage';
 import BurnsidesLemmaPage from './pages/BurnsidesLemmaPage';
+import CubeColoringPage from './pages/CubeColoringPage';
 
 const queryClient = new QueryClient();
 
@@ -83,8 +84,9 @@ const App = () => (
           <Route path="/themes/discrete-math/structures" element={<Structures />} />
           <Route path="/themes/discrete-math/numbers" element={<Numbers />} />
           <Route path="/themes/discrete-math/graphs" element={<Graphs />} />
-          <Route path="/themes/discrete-math/graphs/neighbor-sum-avoidance" element={<NeighborSumAvoidancePage />} />
-          <Route path="/themes/discrete-math/structures/burnsides-lemma" element={<BurnsidesLemmaPage />} />
+           <Route path="/themes/discrete-math/graphs/neighbor-sum-avoidance" element={<NeighborSumAvoidancePage />} />
+           <Route path="/themes/discrete-math/structures/burnsides-lemma" element={<BurnsidesLemmaPage />} />
+           <Route path="/themes/discrete-math/structures/cube-coloring" element={<CubeColoringPage />} />
           <Route path="/themes/social-choice" element={<SocialChoice />} />
           <Route path="/themes/advanced-algorithms" element={<AdvancedAlgorithms />} />
           <Route path="/themes/data-structures" element={<DataStructures />} />
diff --git a/src/components/CubeColoring.tsx b/src/components/CubeColoring.tsx
new file mode 100644
index 0000000..06c468a
--- /dev/null
+++ b/src/components/CubeColoring.tsx
@@ -0,0 +1,345 @@
+import { useState, useRef } from 'react';
+import { Canvas, useFrame } from '@react-three/fiber';
+import { OrbitControls } from '@react-three/drei';
+import { Card, CardContent, CardDescription, CardHeader, CardTitle } from '@/components/ui/card';
+import { Button } from '@/components/ui/button';
+import { Select, SelectContent, SelectItem, SelectTrigger, SelectValue } from '@/components/ui/select';
+import { Badge } from '@/components/ui/badge';
+import SocialShare from './SocialShare';
+import { Box, RotateCw, ArrowRightLeft } from 'lucide-react';
+import * as THREE from 'three';
+
+interface CubeColoringProps {
+  showSocialShare?: boolean;
+  shareUrl?: string;
+}
+
+const colorPalette = [
+  '#ef4444', // red
+  '#3b82f6', // blue
+  '#22c55e', // green
+  '#eab308', // yellow
+  '#a855f7', // purple
+];
+
+// Face indices: 0=top, 1=bottom, 2=front, 3=back, 4=right, 5=left
+const faceNames = ['Top', 'Bottom', 'Front', 'Back', 'Right', 'Left'];
+
+// Rotation definitions: each rotation maps face indices [0,1,2,3,4,5] to new positions
+const rotations = [
+  { name: 'Identity', permutation: [0, 1, 2, 3, 4, 5], description: 'No rotation' },
+  { name: 'Y-90° (vertical)', permutation: [0, 1, 4, 5, 3, 2], description: 'Front→Right→Back→Left' },
+  { name: 'Y-180°', permutation: [0, 1, 3, 2, 5, 4], description: 'Front↔Back, Right↔Left' },
+  { name: 'Y-270°', permutation: [0, 1, 5, 4, 2, 3], description: 'Front→Left→Back→Right' },
+  { name: 'X-90° (horizontal)', permutation: [2, 3, 1, 0, 4, 5], description: 'Top→Front→Bottom→Back' },
+  { name: 'X-180°', permutation: [1, 0, 3, 2, 4, 5], description: 'Top↔Bottom, Front↔Back' },
+  { name: 'X-270°', permutation: [3, 2, 0, 1, 4, 5], description: 'Top→Back→Bottom→Front' },
+  { name: 'Z-90° (front-back)', permutation: [4, 5, 2, 3, 1, 0], description: 'Top→Right→Bottom→Left' },
+  { name: 'Z-180°', permutation: [1, 0, 2, 3, 5, 4], description: 'Top↔Bottom, Right↔Left' },
+  { name: 'Z-270°', permutation: [5, 4, 2, 3, 0, 1], description: 'Top→Left→Bottom→Right' },
+];
+
+interface PaintableCubeProps {
+  colors: string[];
+  onFaceClick: (faceIndex: number) => void;
+}
+
+function PaintableCube({ colors, onFaceClick }: PaintableCubeProps) {
+  const meshRef = useRef<THREE.Mesh>(null);
+
+  useFrame(() => {
+    if (meshRef.current) {
+      meshRef.current.rotation.y += 0.005;
+    }
+  });
+
+  return (
+    <mesh ref={meshRef}>
+      <boxGeometry args={[2, 2, 2]} />
+      {colors.map((color, i) => (
+        <meshStandardMaterial key={i} attach={`material-${i}`} color={color} />
+      ))}
+    </mesh>
+  );
+}
+
+function PermutationCube({ permutation }: { permutation: number[] }) {
+  const meshRef = useRef<THREE.Mesh>(null);
+
+  useFrame(() => {
+    if (meshRef.current) {
+      meshRef.current.rotation.y += 0.005;
+    }
+  });
+
+  // Create materials based on permutation
+  const baseMaterials = [
+    new THREE.MeshStandardMaterial({ color: '#ef4444' }), // red - top
+    new THREE.MeshStandardMaterial({ color: '#3b82f6' }), // blue - bottom
+    new THREE.MeshStandardMaterial({ color: '#22c55e' }), // green - front
+    new THREE.MeshStandardMaterial({ color: '#eab308' }), // yellow - back
+    new THREE.MeshStandardMaterial({ color: '#a855f7' }), // purple - right
+    new THREE.MeshStandardMaterial({ color: '#f97316' }), // orange - left
+  ];
+
+  // Apply permutation: materials[i] shows the color that should be at position i
+  const permutedMaterials = permutation.map(originalIndex => baseMaterials[originalIndex]);
+
+  return (
+    <mesh ref={meshRef}>
+      <boxGeometry args={[2, 2, 2]} />
+      {permutedMaterials.map((material, i) => (
+        <primitive key={i} object={material} attach={`material-${i}`} />
+      ))}
+    </mesh>
+  );
+}
+
+const CubeColoring = ({ showSocialShare = false, shareUrl }: CubeColoringProps) => {
+  const [faceColors, setFaceColors] = useState<string[]>(Array(6).fill(colorPalette[0]));
+  const [selectedColor, setSelectedColor] = useState(colorPalette[0]);
+  const [selectedPermutation, setSelectedPermutation] = useState<number[]>([0, 1, 2, 3, 4, 5]);
+  const [selectedRotation, setSelectedRotation] = useState(0);
+
+  const handleFaceClick = (faceIndex: number) => {
+    const newColors = [...faceColors];
+    newColors[faceIndex] = selectedColor;
+    setFaceColors(newColors);
+  };
+
+  const handlePermutationChange = (position: number, value: string) => {
+    const newPerm = [...selectedPermutation];
+    newPerm[position] = parseInt(value);
+    setSelectedPermutation(newPerm);
+  };
+
+  const permutationToRotation = () => {
+    const match = rotations.findIndex(r => 
+      r.permutation.every((val, idx) => val === selectedPermutation[idx])
+    );
+    return match >= 0 ? rotations[match] : null;
+  };
+
+  return (
+    <div className="space-y-6">
+      <Card>
+        <CardHeader>
+          <CardTitle className="flex items-center gap-2">
+            <Box className="w-6 h-6" />
+            Burnside's Lemma: Cube Coloring
+          </CardTitle>
+          <CardDescription>
+            How many distinct cubes can you make by painting each face in one of 5 colors?
+            Two cubes are the same if one can be rotated to look like the other.
+          </CardDescription>
+        </CardHeader>
+        <CardContent className="space-y-8">
+          {/* 3D Cube Arena */}
+          <div className="space-y-4">
+            <h3 className="text-lg font-semibold">Paint Your Cube</h3>
+            <div className="grid md:grid-cols-2 gap-6">
+              <div className="h-[300px] border border-border rounded-lg bg-muted/20">
+                <Canvas camera={{ position: [4, 4, 4] }}>
+                  <ambientLight intensity={0.5} />
+                  <directionalLight position={[10, 10, 5]} intensity={1} />
+                  <PaintableCube colors={faceColors} onFaceClick={handleFaceClick} />
+                  <OrbitControls enableZoom={true} />
+                </Canvas>
+              </div>
+              
+              <div className="space-y-4">
+                <div>
+                  <label className="text-sm font-medium mb-2 block">Select Color</label>
+                  <div className="flex gap-2">
+                    {colorPalette.map((color) => (
+                      <button
+                        key={color}
+                        onClick={() => setSelectedColor(color)}
+                        className={`w-12 h-12 rounded-lg border-2 transition-all ${
+                          selectedColor === color ? 'border-foreground scale-110' : 'border-border'
+                        }`}
+                        style={{ backgroundColor: color }}
+                      />
+                    ))}
+                  </div>
+                </div>
+                
+                <div>
+                  <label className="text-sm font-medium mb-2 block">Face Colors</label>
+                  <div className="grid grid-cols-2 gap-2">
+                    {faceNames.map((name, idx) => (
+                      <div key={idx} className="flex items-center gap-2 p-2 bg-muted/50 rounded">
+                        <div
+                          className="w-6 h-6 rounded border border-border"
+                          style={{ backgroundColor: faceColors[idx] }}
+                        />
+                        <span className="text-sm">{name}</span>
+                      </div>
+                    ))}
+                  </div>
+                </div>
+              </div>
+            </div>
+          </div>
+
+          {/* Permutation to Rotation */}
+          <div className="space-y-4">
+            <h3 className="text-lg font-semibold flex items-center gap-2">
+              <ArrowRightLeft className="w-5 h-5" />
+              Permutation → Rotation
+            </h3>
+            <div className="grid md:grid-cols-2 gap-6">
+              <Card className="bg-muted/30">
+                <CardHeader>
+                  <CardTitle className="text-base">Pick a Permutation</CardTitle>
+                  <CardDescription className="text-xs">
+                    Choose where each face (0-5) goes
+                  </CardDescription>
+                </CardHeader>
+                <CardContent className="space-y-3">
+                  {selectedPermutation.map((value, idx) => (
+                    <div key={idx} className="flex items-center gap-3">
+                      <Badge variant="outline" className="w-20">
+                        Face {idx}
+                      </Badge>
+                      <span className="text-muted-foreground">→</span>
+                      <Select
+                        value={value.toString()}
+                        onValueChange={(v) => handlePermutationChange(idx, v)}
+                      >
+                        <SelectTrigger className="w-24">
+                          <SelectValue />
+                        </SelectTrigger>
+                        <SelectContent>
+                          {[0, 1, 2, 3, 4, 5].map((n) => (
+                            <SelectItem key={n} value={n.toString()}>
+                              {n}
+                            </SelectItem>
+                          ))}
+                        </SelectContent>
+                      </Select>
+                      <span className="text-xs text-muted-foreground">({faceNames[idx]})</span>
+                    </div>
+                  ))}
+                </CardContent>
+              </Card>
+
+              <Card className="bg-muted/30">
+                <CardHeader>
+                  <CardTitle className="text-base">Corresponding Rotation</CardTitle>
+                  <CardDescription className="text-xs">
+                    Visualize the rotation
+                  </CardDescription>
+                </CardHeader>
+                <CardContent className="space-y-4">
+                  {permutationToRotation() ? (
+                    <>
+                      <div className="text-center space-y-2">
+                        <Badge className="text-lg px-4 py-2">
+                          {permutationToRotation()!.name}
+                        </Badge>
+                        <p className="text-sm text-muted-foreground">
+                          {permutationToRotation()!.description}
+                        </p>
+                      </div>
+                      <div className="h-[200px] border border-border rounded-lg bg-background/50">
+                        <Canvas camera={{ position: [4, 4, 4] }}>
+                          <ambientLight intensity={0.5} />
+                          <directionalLight position={[10, 10, 5]} intensity={1} />
+                          <PermutationCube permutation={selectedPermutation} />
+                          <OrbitControls enableZoom={true} />
+                        </Canvas>
+                      </div>
+                    </>
+                  ) : (
+                    <div className="text-center py-8 text-muted-foreground">
+                      <p>Invalid permutation</p>
+                      <p className="text-xs mt-1">Each number 0-5 must appear exactly once</p>
+                    </div>
+                  )}
+                </CardContent>
+              </Card>
+            </div>
+          </div>
+
+          {/* Rotation to Permutation */}
+          <div className="space-y-4">
+            <h3 className="text-lg font-semibold flex items-center gap-2">
+              <RotateCw className="w-5 h-5" />
+              Rotation → Permutation
+            </h3>
+            <div className="grid md:grid-cols-2 gap-6">
+              <Card className="bg-muted/30">
+                <CardHeader>
+                  <CardTitle className="text-base">Corresponding Permutation</CardTitle>
+                  <CardDescription className="text-xs">
+                    How faces move with this rotation
+                  </CardDescription>
+                </CardHeader>
+                <CardContent className="space-y-3">
+                  {rotations[selectedRotation].permutation.map((target, source) => (
+                    <div key={source} className="flex items-center gap-3">
+                      <Badge variant="outline" className="w-20">
+                        Face {source}
+                      </Badge>
+                      <span className="text-muted-foreground">→</span>
+                      <Badge className="w-20">Face {target}</Badge>
+                      <span className="text-xs text-muted-foreground">
+                        {faceNames[source]} → {faceNames[target]}
+                      </span>
+                    </div>
+                  ))}
+                </CardContent>
+              </Card>
+
+              <Card className="bg-muted/30">
+                <CardHeader>
+                  <CardTitle className="text-base">Pick a Rotation</CardTitle>
+                  <CardDescription className="text-xs">
+                    Select a cube rotation
+                  </CardDescription>
+                </CardHeader>
+                <CardContent className="space-y-4">
+                  <Select
+                    value={selectedRotation.toString()}
+                    onValueChange={(v) => setSelectedRotation(parseInt(v))}
+                  >
+                    <SelectTrigger>
+                      <SelectValue />
+                    </SelectTrigger>
+                    <SelectContent>
+                      {rotations.map((rot, idx) => (
+                        <SelectItem key={idx} value={idx.toString()}>
+                          {rot.name} - {rot.description}
+                        </SelectItem>
+                      ))}
+                    </SelectContent>
+                  </Select>
+
+                  <div className="h-[200px] border border-border rounded-lg bg-background/50">
+                    <Canvas camera={{ position: [4, 4, 4] }}>
+                      <ambientLight intensity={0.5} />
+                      <directionalLight position={[10, 10, 5]} intensity={1} />
+                      <PermutationCube permutation={rotations[selectedRotation].permutation} />
+                      <OrbitControls enableZoom={true} />
+                    </Canvas>
+                  </div>
+                </CardContent>
+              </Card>
+            </div>
+          </div>
+        </CardContent>
+      </Card>
+
+      {showSocialShare && (
+        <SocialShare
+          url={shareUrl || window.location.href}
+          title="Burnside's Lemma - Cube Coloring"
+          description="Explore cube rotations and permutations with Burnside's Lemma"
+        />
+      )}
+    </div>
+  );
+};
+
+export default CubeColoring;
diff --git a/src/components/InteractiveIndex.tsx b/src/components/InteractiveIndex.tsx
index b25fd91..7e75219 100644
--- a/src/components/InteractiveIndex.tsx
+++ b/src/components/InteractiveIndex.tsx
@@ -261,6 +261,15 @@ const allInteractives: Interactive[] = [
     path: '/themes/discrete-math/structures/burnsides-lemma',
     theme: 'Discrete Math',
     dateAdded: '2025-01-22'
+  },
+  {
+    id: 'cube-coloring',
+    title: "Burnside's Lemma: Cube Coloring",
+    description: 'Explore cube rotations and their correspondence with permutations. Paint a 3D cube and understand the rotational symmetry group.',
+    tags: ['group-theory', 'symmetry', 'cube', 'rotations', 'permutations', 'algebra', '3d'],
+    path: '/themes/discrete-math/structures/cube-coloring',
+    theme: 'Discrete Math',
+    dateAdded: '2025-01-22'
   }
 ];
 
diff --git a/src/pages/CubeColoringPage.tsx b/src/pages/CubeColoringPage.tsx
new file mode 100644
index 0000000..30ddb00
--- /dev/null
+++ b/src/pages/CubeColoringPage.tsx
@@ -0,0 +1,17 @@
+import Layout from '@/components/Layout';
+import CubeColoring from '@/components/CubeColoring';
+
+const CubeColoringPage = () => {
+  return (
+    <Layout>
+      <div className="container mx-auto px-4 py-8">
+        <CubeColoring 
+          showSocialShare={true}
+          shareUrl={window.location.href}
+        />
+      </div>
+    </Layout>
+  );
+};
+
+export default CubeColoringPage;
diff --git a/src/pages/theme-pages/discrete-math/Structures.tsx b/src/pages/theme-pages/discrete-math/Structures.tsx
index be94b99..ac39aa3 100644
--- a/src/pages/theme-pages/discrete-math/Structures.tsx
+++ b/src/pages/theme-pages/discrete-math/Structures.tsx
@@ -5,6 +5,7 @@ import { Badge } from '@/components/ui/badge';
 import Layout from '@/components/Layout';
 import { useState } from 'react';
 import BurnsidesLemma from '@/components/BurnsidesLemma';
+import CubeColoring from '@/components/CubeColoring';
 
 interface Interactive {
   id: string;
@@ -24,6 +25,14 @@ const interactives: Interactive[] = [
     path: '/themes/discrete-math/structures/burnsides-lemma',
     component: BurnsidesLemma
   },
+  {
+    id: 'cube-coloring',
+    title: "Burnside's Lemma: Cube Coloring",
+    description: 'Explore cube rotations and their correspondence with permutations. Paint a 3D cube and understand the rotational symmetry group.',
+    tags: ['group-theory', 'symmetry', 'cube', 'rotations', 'permutations', 'algebra', '3d'],
+    path: '/themes/discrete-math/structures/cube-coloring',
+    component: CubeColoring
+  },
 ];
 
 const Structures = () => {