DanAllan.com

curiosity in science, music, and media

Uno Games

August 11th, 2008 · 2 Comments

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

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

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.)

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Coming Back Online

August 11th, 2008 · No Comments

The creations (pictures and movies) and many more recent projects are coming back online in the next few days.

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EHS Band Photo Tile

August 25th, 2004 · No Comments

This collection of unaltered photographs from band events assembles into a picture of my high school band director (Mr. Borsz!!!) conducting on the marching band field.

Click on the photo to see a more detailed view.

(This was assembled using freeware called Phototile. It is no longer available from its official website, but may be found elsewhere.)

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Cameo

April 18th, 2004 · No Comments

In these two 10-minute features, Andy Moss and I barge in on some of the greatest moments in cinema history, replacing the actors who were actually there. (Andy is now a professional actor. Between his charisma and my most ambitious visual effects yet, I think this is the most fun of all the videos on the site.)

Cameo: Part I


Cameo: Part II


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Track Motiviation

November 8th, 2003 · No Comments

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Old Man Borsz

October 7th, 2003 · No Comments

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One Ring

September 10th, 2003 · No Comments

A close look reveals the source of Mr. Hurd’s malevolence.

“The Ring of Power corrupts the hearts of men. Even kings history teachers cannot withstand its power.” - Gandalf the Grey.

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High School Videos

May 14th, 2003 · 1 Comment

Purely inside jokes, preserved here for sentimental reasons.

Star Wars Toys


Borsz in Bondage


The Quest for Cheesecake


Boomwhackers!


Banana Milk


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Oldmember

March 25th, 2003 · No Comments

Mr. Ruocco/Powers was the victim of a school fundraiser. Heather Graham rewards him for his good humor.

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Breaking Bobby-O

November 18th, 2002 · No Comments

In a scene from The Man Who Came to Dinner, one of the actors thumbs through some photographs. We in the cast tried to make him crack a smile during the performances, slipping in prank photos. These are highlights.

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