fix: always show game control buttons in room-based sessions

In room-based multiplayer games, force canModifyPlayers=false to ensure
Setup, New Game, and Quit buttons are always visible to all room members.

The arcade session locking mechanism (hasActiveSession) doesn't apply
to room-based games, so we detect rooms via isInRoom and override the
canModifyPlayers logic.

This fixes the issue where some room members saw buttons while others
didn't, depending on their arcade session state.

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

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

apps/web/src/app/arcade/matching/components/MemoryPairsGame.tsx

@@ -4,6 +4,7 @@ import { useRouter } from 'next/navigation'
 import { useEffect, useRef } from 'react'
 import { PageWithNav } from '@/components/PageWithNav'
 import { useArcadeRedirect } from '@/hooks/useArcadeRedirect'
+import { useRoomData } from '@/hooks/useRoomData'
 import { css } from '../../../../../styled-system/css'
 import { StandardGameLayout } from '../../../../components/StandardGameLayout'
 import { useFullscreen } from '../../../../contexts/FullscreenContext'
@@ -14,9 +15,13 @@ import { SetupPhase } from './SetupPhase'
 
 export function MemoryPairsGame() {
   const router = useRouter()
-  const { state, exitSession, resetGame } = useMemoryPairs()
+  const { state, exitSession, resetGame, goToSetup } = useMemoryPairs()
   const { setFullscreenElement } = useFullscreen()
-  const { canModifyPlayers } = useArcadeRedirect({ currentGame: 'matching' })
+  const { isInRoom } = useRoomData()
+  const arcadeRedirect = useArcadeRedirect({ currentGame: 'matching' })
+  // In rooms, always show buttons (canModifyPlayers = false shows buttons)
+  // In arcade sessions, use normal arcade redirect logic
+  const canModifyPlayers = isInRoom ? false : arcadeRedirect.canModifyPlayers
   const gameRef = useRef<HTMLDivElement>(null)
 
   useEffect(() => {
@@ -37,11 +42,14 @@ export function MemoryPairsGame() {
         exitSession()
         router.push('/arcade')
       }}
-      onSetup={() => {
-        // Exit current session and return to arcade (which will redirect to setup)
-        exitSession()
-        router.push('/arcade/matching')
-      }}
+      onSetup={
+        goToSetup
+          ? () => {
+              // Transition to setup phase (will pause game if active)
+              goToSetup()
+            }
+          : undefined
+      }
       onNewGame={() => {
         resetGame()
       }}