feat(arcade): register Math Sprint in game system

Register math-sprint in all required places:
- validators.ts: Add mathSprintValidator to registry
- game-registry.ts: Register mathSprintGame
- game-configs.ts: Add MathSprintGameConfig type and defaults
- game-config-helpers.ts: Add config getters and validation

This demonstrates the boilerplate issue documented in AUDIT_2:
Had to update 4 files with switch/case statements and type definitions.

Addresses issue #2 and #3 from architecture audit.

apps/web/src/lib/arcade/game-config-helpers.ts

@@ -15,6 +15,7 @@ import {
   DEFAULT_MEMORY_QUIZ_CONFIG,
   DEFAULT_COMPLEMENT_RACE_CONFIG,
   DEFAULT_NUMBER_GUESSER_CONFIG,
+  DEFAULT_MATH_SPRINT_CONFIG,
 } from './game-configs'
 
 /**
@@ -36,6 +37,8 @@ function getDefaultGameConfig(gameName: ExtendedGameName): GameConfigByName[Exte
       return DEFAULT_COMPLEMENT_RACE_CONFIG
     case 'number-guesser':
       return DEFAULT_NUMBER_GUESSER_CONFIG
+    case 'math-sprint':
+      return DEFAULT_MATH_SPRINT_CONFIG
     default:
       throw new Error(`Unknown game: ${gameName}`)
   }
@@ -215,6 +218,18 @@ export function validateGameConfig(gameName: ExtendedGameName, config: any): boo
         config.roundsToWin >= 1
       )
 
+    case 'math-sprint':
+      return (
+        typeof config === 'object' &&
+        config !== null &&
+        ['easy', 'medium', 'hard'].includes(config.difficulty) &&
+        typeof config.questionsPerRound === 'number' &&
+        typeof config.timePerQuestion === 'number' &&
+        config.questionsPerRound >= 5 &&
+        config.questionsPerRound <= 20 &&
+        config.timePerQuestion >= 10
+      )
+
     default:
       return false
   }

apps/web/src/lib/arcade/game-configs.ts

@@ -11,6 +11,7 @@
 
 import type { DifficultyLevel } from '@/app/arcade/memory-quiz/types'
 import type { Difficulty, GameType } from '@/app/games/matching/context/types'
+import type { Difficulty as MathSprintDifficulty } from '@/arcade-games/math-sprint/types'
 
 /**
  * Configuration for matching (memory pairs) game
@@ -49,6 +50,15 @@ export interface NumberGuesserGameConfig {
   roundsToWin: number
 }
 
+/**
+ * Configuration for math-sprint game
+ */
+export interface MathSprintGameConfig {
+  difficulty: MathSprintDifficulty
+  questionsPerRound: number
+  timePerQuestion: number
+}
+
 /**
  * Union type of all game configs for type-safe access
  */
@@ -57,6 +67,7 @@ export type GameConfigByName = {
   'memory-quiz': MemoryQuizGameConfig
   'complement-race': ComplementRaceGameConfig
   'number-guesser': NumberGuesserGameConfig
+  'math-sprint': MathSprintGameConfig
 }
 
 /**
@@ -68,6 +79,7 @@ export interface RoomGameConfig {
   'memory-quiz'?: MemoryQuizGameConfig
   'complement-race'?: ComplementRaceGameConfig
   'number-guesser'?: NumberGuesserGameConfig
+  'math-sprint'?: MathSprintGameConfig
 }
 
 /**
@@ -95,3 +107,9 @@ export const DEFAULT_NUMBER_GUESSER_CONFIG: NumberGuesserGameConfig = {
   maxNumber: 100,
   roundsToWin: 3,
 }
+
+export const DEFAULT_MATH_SPRINT_CONFIG: MathSprintGameConfig = {
+  difficulty: 'medium',
+  questionsPerRound: 10,
+  timePerQuestion: 30,
+}

apps/web/src/lib/arcade/game-registry.ts

@@ -107,5 +107,7 @@ export function clearRegistry(): void {
 // ============================================================================
 
 import { numberGuesserGame } from '@/arcade-games/number-guesser'
+import { mathSprintGame } from '@/arcade-games/math-sprint'
 
 registerGame(numberGuesserGame)
+registerGame(mathSprintGame)

apps/web/src/lib/arcade/validators.ts

@@ -13,6 +13,7 @@
 import { matchingGameValidator } from './validation/MatchingGameValidator'
 import { memoryQuizGameValidator } from './validation/MemoryQuizGameValidator'
 import { numberGuesserValidator } from '@/arcade-games/number-guesser/Validator'
+import { mathSprintValidator } from '@/arcade-games/math-sprint/Validator'
 import type { GameValidator } from './validation/types'
 
 /**
@@ -24,6 +25,7 @@ export const validatorRegistry = {
   matching: matchingGameValidator,
   'memory-quiz': memoryQuizGameValidator,
   'number-guesser': numberGuesserValidator,
+  'math-sprint': mathSprintValidator,
   // Add new games here - GameName type will auto-update
 } as const
 
@@ -65,4 +67,9 @@ export function getRegisteredGameNames(): GameName[] {
 /**
  * Re-export validators for backwards compatibility
  */
-export { matchingGameValidator, memoryQuizGameValidator, numberGuesserValidator }
+export {
+  matchingGameValidator,
+  memoryQuizGameValidator,
+  numberGuesserValidator,
+  mathSprintValidator,
+}