diff --git a/src/components/DominoRetilingPuzzle.tsx b/src/components/DominoRetilingPuzzle.tsx
index 65898b3..b58aa02 100644
--- a/src/components/DominoRetilingPuzzle.tsx
+++ b/src/components/DominoRetilingPuzzle.tsx
@@ -28,44 +28,62 @@ const DominoRetilingPuzzle: React.FC<DominoRetilingPuzzleProps> = ({ showSocialS
   const BOARD_ROWS = boardSize;
   const BOARD_COLS = boardSize + 1; // n + 1 extra square on top row
 
-  // Generate a random valid tiling of the n×n board
+  // Generate a random valid tiling of the n×n board using backtracking
   const generateRandomTiling = useCallback(() => {
     const newDominoes: Domino[] = [];
     const occupied = new Set<string>();
     let dominoId = 0;
 
-    // Fill the n×n board with random dominoes
-    for (let row = 0; row < boardSize; row++) {
-      for (let col = 0; col < boardSize; col++) {
-        const key = `${row},${col}`;
-        if (occupied.has(key)) continue;
-
-        // Randomly choose horizontal or vertical
-        const canHorizontal = col + 1 < boardSize && !occupied.has(`${row},${col + 1}`);
-        const canVertical = row + 1 < boardSize && !occupied.has(`${row + 1},${col}`);
-
-        if (!canHorizontal && !canVertical) continue;
-
-        let isHorizontal: boolean;
-        if (canHorizontal && canVertical) {
-          isHorizontal = Math.random() > 0.5;
-        } else {
-          isHorizontal = canHorizontal;
+    // Helper to find first uncovered cell
+    const findFirstUncovered = (): { row: number; col: number } | null => {
+      for (let row = 0; row < boardSize; row++) {
+        for (let col = 0; col < boardSize; col++) {
+          if (!occupied.has(`${row},${col}`)) {
+            return { row, col };
+          }
         }
+      }
+      return null;
+    };
 
-        newDominoes.push({
-          id: dominoId++,
-          row,
-          col,
-          isHorizontal
-        });
+    // Greedy fill with backtracking if needed
+    while (true) {
+      const cell = findFirstUncovered();
+      if (!cell) break; // All covered
 
-        occupied.add(key);
-        if (isHorizontal) {
-          occupied.add(`${row},${col + 1}`);
-        } else {
-          occupied.add(`${row + 1},${col}`);
-        }
+      const { row, col } = cell;
+      
+      // Try to place a domino here
+      const canHorizontal = col + 1 < boardSize && !occupied.has(`${row},${col + 1}`);
+      const canVertical = row + 1 < boardSize && !occupied.has(`${row + 1},${col}`);
+
+      if (!canHorizontal && !canVertical) {
+        // Stuck - this shouldn't happen with even board sizes, but reset if it does
+        newDominoes.length = 0;
+        occupied.clear();
+        dominoId = 0;
+        continue;
+      }
+
+      let isHorizontal: boolean;
+      if (canHorizontal && canVertical) {
+        isHorizontal = Math.random() > 0.5;
+      } else {
+        isHorizontal = canHorizontal;
+      }
+
+      newDominoes.push({
+        id: dominoId++,
+        row,
+        col,
+        isHorizontal
+      });
+
+      occupied.add(`${row},${col}`);
+      if (isHorizontal) {
+        occupied.add(`${row},${col + 1}`);
+      } else {
+        occupied.add(`${row + 1},${col}`);
       }
     }
 
@@ -174,6 +192,25 @@ const DominoRetilingPuzzle: React.FC<DominoRetilingPuzzleProps> = ({ showSocialS
   const renderBoard = () => {
     const cells: JSX.Element[] = [];
 
+    // Render the inaccessible shaded region (last column except top cell)
+    for (let row = 1; row < BOARD_ROWS; row++) {
+      cells.push(
+        <div
+          key={`inaccessible-${row}`}
+          className="border border-border/30"
+          style={{
+            position: 'absolute',
+            left: boardSize * cellSize,
+            top: row * cellSize,
+            width: cellSize,
+            height: cellSize,
+            backgroundColor: 'rgba(100, 100, 100, 0.3)',
+            backgroundImage: 'repeating-linear-gradient(45deg, transparent, transparent 4px, rgba(80, 80, 80, 0.4) 4px, rgba(80, 80, 80, 0.4) 8px)',
+          }}
+        />
+      );
+    }
+
     // Render cells
     for (let row = 0; row < BOARD_ROWS; row++) {
       const maxCol = row === 0 ? BOARD_COLS : boardSize;
@@ -193,15 +230,10 @@ const DominoRetilingPuzzle: React.FC<DominoRetilingPuzzleProps> = ({ showSocialS
             key={`cell-${row}-${col}`}
             className={`border border-border/50 ${
               isExtraSquare 
-                ? 'bg-blue-500/40' 
+                ? 'bg-blue-400/50' 
                 : (row + col) % 2 === 0 ? 'bg-muted/30' : 'bg-muted/60'
             } ${selectedDomino !== null ? 'cursor-pointer' : ''}`}
-            style={{
-              ...cellStyle,
-              ...(isExtraSquare ? {
-                backgroundImage: 'repeating-linear-gradient(45deg, transparent, transparent 4px, rgba(59, 130, 246, 0.3) 4px, rgba(59, 130, 246, 0.3) 8px)',
-              } : {})
-            }}
+            style={cellStyle}
             onMouseEnter={() => handleCellHover(row, col)}
             onMouseLeave={() => setHoverPlacement(null)}
             onClick={() => {