fix(memory-quiz): scope game settings by game name for proper persistence

Previously, settings were stored at the root of gameConfig, causing each
game to overwrite the other's settings when switching between games.

Now settings are stored under gameConfig['memory-quiz'], allowing each
game to maintain its own settings independently. When you switch from
memory-quiz to another game and back, the memory-quiz settings are
preserved exactly as you left them.

Changes:
- Load settings from gameConfig['memory-quiz'] instead of root gameConfig
- Save settings to gameConfig['memory-quiz'] to avoid overwriting other games
- Added comments explaining the scoping strategy

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

apps/web/src/app/arcade/memory-quiz/context/RoomMemoryQuizProvider.tsx

@@ -248,11 +248,16 @@ export function RoomMemoryQuizProvider({ children }: { children: ReactNode }) {
   const [localCurrentInput, setLocalCurrentInput] = useState('')
 
   // Merge saved game config from room with initialState
+  // Settings are scoped by game name to preserve settings when switching games
   const mergedInitialState = useMemo(() => {
-    const savedConfig = roomData?.gameConfig as Record<string, any> | null | undefined
+    const gameConfig = roomData?.gameConfig as Record<string, any> | null | undefined
+    if (!gameConfig) return initialState
+
+    // Get settings for this specific game (memory-quiz)
+    const savedConfig = gameConfig['memory-quiz'] as Record<string, any> | null | undefined
     if (!savedConfig) return initialState
 
-    console.log('[RoomMemoryQuizProvider] Loading saved game config:', savedConfig)
+    console.log('[RoomMemoryQuizProvider] Loading saved game config for memory-quiz:', savedConfig)
 
     return {
       ...initialState,
@@ -441,12 +446,20 @@ export function RoomMemoryQuizProvider({ children }: { children: ReactNode }) {
       })
 
       // Save setting to room's gameConfig for persistence
+      // Settings are scoped by game name to preserve settings when switching games
       if (roomData?.id) {
+        const currentGameConfig = (roomData.gameConfig as Record<string, any>) || {}
+        const currentMemoryQuizConfig =
+          (currentGameConfig['memory-quiz'] as Record<string, any>) || {}
+
         const updatedConfig = {
-          ...(roomData.gameConfig as Record<string, any>),
-          [field]: value,
+          ...currentGameConfig,
+          'memory-quiz': {
+            ...currentMemoryQuizConfig,
+            [field]: value,
+          },
         }
-        console.log('[RoomMemoryQuizProvider] Saving game config:', updatedConfig)
+        console.log('[RoomMemoryQuizProvider] Saving game config for memory-quiz:', updatedConfig)
         updateGameConfig({
           roomId: roomData.id,
           gameConfig: updatedConfig,