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/types.go

@@ -14,6 +14,13 @@ const (
 	DirRight
 )
 
+type GameMode int
+
+const (
+	ModeMultiplayer  GameMode = iota // Default (0) - backward compatible
+	ModeSinglePlayer                 // Single player survival mode
+)
+
 // Opposite returns true if a and b are 180-degree reversals.
 func (d Direction) Opposite(other Direction) bool {
 	switch d {
@@ -88,6 +95,8 @@ type SnakeGame struct {
 	Winner        *Player   // nil if draw
 	CountdownEnd  time.Time // when countdown reaches 0
 	RematchGameID *string
+	Mode          GameMode // ModeMultiplayer or ModeSinglePlayer
+	Score         int      // tracks food eaten in single player
 }
 
 func (sg *SnakeGame) IsFinished() bool {
@@ -132,6 +141,7 @@ func (sg *SnakeGame) snapshot() *SnakeGame {
 	}
 	cp.Players = make([]*Player, len(sg.Players))
 	copy(cp.Players, sg.Players)
+	// Mode and Score are value types, already copied by *sg
 	return &cp
 }