feat: add single player snake mode

Add solo mode where players survive as long as possible while tracking
score (food eaten). Single player games start with a shorter 3-second
countdown vs 10 seconds for multiplayer, maintain exactly 1 food item
for classic snake feel, and end when the player dies rather than when
one player remains.

- Add GameMode type (ModeMultiplayer/ModeSinglePlayer) and Score field
- Filter single player games from "Join a Game" lobby list
- Show "Ready?" and "Score: X" UI for single player mode
- Hide invite link for single player games
- Preserve game mode on rematch

snake/loop.go

@@ -5,17 +5,23 @@ import (
 )
 
 const (
-	targetFPS        = 60
-	tickInterval     = time.Second / targetFPS
-	snakeSpeed       = 7 // cells per second
-	moveInterval     = time.Second / snakeSpeed
-	countdownSeconds = 10
-	inactivityLimit  = 60 * time.Second
+	targetFPS                      = 60
+	tickInterval                   = time.Second / targetFPS
+	snakeSpeed                     = 7 // cells per second
+	moveInterval                   = time.Second / snakeSpeed
+	countdownSecondsMultiplayer    = 10
+	countdownSecondsSinglePlayer   = 3
+	inactivityLimit                = 60 * time.Second
 )
 
 func (si *SnakeGameInstance) startOrResetCountdownLocked() {
 	si.game.Status = StatusCountdown
-	si.game.CountdownEnd = time.Now().Add(countdownSeconds * time.Second)
+
+	countdown := countdownSecondsMultiplayer
+	if si.game.Mode == ModeSinglePlayer {
+		countdown = countdownSecondsSinglePlayer
+	}
+	si.game.CountdownEnd = time.Now().Add(time.Duration(countdown) * time.Second)
 
 	si.loopOnce.Do(func() {
 		si.stopCh = make(chan struct{})
@@ -86,7 +92,7 @@ func (si *SnakeGameInstance) initGame() {
 
 	state := si.game.State
 	state.Snakes = SpawnSnakes(activeSlots, state.Width, state.Height)
-	SpawnFood(state, len(activeSlots))
+	SpawnFood(state, len(activeSlots), si.game.Mode)
 }
 
 func (si *SnakeGameInstance) gamePhase() {
@@ -152,16 +158,33 @@ func (si *SnakeGameInstance) gamePhase() {
 			// Check food eaten (only by surviving snakes)
 			eaten := CheckFood(state)
 			RemoveFood(state, eaten)
-			SpawnFood(state, si.game.PlayerCount())
+			SpawnFood(state, si.game.PlayerCount(), si.game.Mode)
+
+			// Track score for single player
+			si.game.Score += len(eaten)
 
 			// Check game over
 			alive := AliveCount(state)
-			if alive <= 1 {
-				si.game.Status = StatusFinished
-				winnerIdx := LastAlive(state)
-				if winnerIdx >= 0 && winnerIdx < len(si.game.Players) {
-					si.game.Winner = si.game.Players[winnerIdx]
+			gameOver := false
+			if si.game.Mode == ModeSinglePlayer {
+				// Single player ends when the player dies (alive == 0)
+				if alive == 0 {
+					gameOver = true
+					// No winner in single player - just final score
 				}
+			} else {
+				// Multiplayer ends when 1 or fewer alive
+				if alive <= 1 {
+					gameOver = true
+					winnerIdx := LastAlive(state)
+					if winnerIdx >= 0 && winnerIdx < len(si.game.Players) {
+						si.game.Winner = si.game.Players[winnerIdx]
+					}
+				}
+			}
+
+			if gameOver {
+				si.game.Status = StatusFinished
 			}
 
 			if si.persister != nil {
@@ -171,7 +194,7 @@ func (si *SnakeGameInstance) gamePhase() {
 			si.gameMu.Unlock()
 			si.notify()
 
-			if alive <= 1 {
+			if gameOver {
 				return
 			}
 		}