Uno games can go on forever. Game length depends mainly on how the cards get shuffled. How long is an Uno game likely to take? Does including more players make a longer game or just more chances for someone to win and end it?
Sounds like the perfect way to squirm out of taking History of Mathematics by writing an independent paper about Uno! My musings here are brief and not math-y.
Most people play Uno with a pretty basic strategy, and it’s not hard to program a computer to play as well as a casual human player. My laptop can simulate one million Uno games in about five minutes. Then it tells me how long the games took. The results look like this.
Red = 2-player games; Green = 3-player games; Blue = 4-player games
See that the blue line generally stays higher than the red line? This tells us that adding players makes the game take longer. Not a surprise.
But there are two ways to count how long an Uno game takes. You could count total number of turns or you could count rounds — that is, times that play circles around the table. (Rounds = Turns / # Players.) Here’s the same data, counting rounds this time.
Red = 2-player games; Green = 3-player games; Blue = 4-player games
The curves have changed places: adding players makes for shorter Uno games if we count rounds. This makes sense: more players means more chances that someone is holding a winning hand.
The Conclusion: Uno games with more players tend to take longer in terms of actual turns, but it takes fewer rounds of play for someone to win.
Another Quick Observation: The first player has a measurable advantage. Each player has (approximately) a 0.5% better chance of winning than the player who plays after him. (This is true no matter how many people are playing.) Obviously, the advantage is small. Practically speaking, it makes little difference who goes first.
Extra Credit
- I only showed plots for 2-4 players, but the trend continues. I tested up to 12-player games.
- The first peak, at 7 rounds, corresponds to Player 1 holding a perfect hand, going out at the earliest possible turn. The secondary peak, at 11 rounds, corresponds to the same scenario but with Player 1 having been hit with a Draw 4 card. This can be confirmed by simulating games using a deck without Draw 4 cards in it.
Red = standard deck; Blue = deck without Draw 4 or Wild Draw 4 cards
- Varying the players’ strategies has a low impact on game length.
If you have ideas, I’d be happy to hear them. I will share my Uno-playing code upon request. (It’s written in neatly commented Java.)
