This sounds like the perfect way to squirm out of a college course requirement by writing an independent paper about Uno!

Most people play Uno with a casual strategy, and it’s not hard to program a computer to imitate that. 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

First Conclusion: 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 / Number of Players.) Here’s the same data, counted in rounds.

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.

Going First Helps. The first player has a small but measurable advantage. In fact, each player has a 0.5% better chance of winning than the player who plays after him. (This works for big games, too. I ran simulations up to 12 players.)

What’s with those peaks? 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 Player 1 was also hit by 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 card