I considered two questions about the board game Risk.

1. What are the odds of winning a particular dice roll? (like 3 dice vs. 2 dice, 3 dice vs. 1 die, etc.)
2. What are the odds of conquering a territory? (for example, 20 men invading 17 men)

Anyone who has played risk has an intuitive sense of the answers to the first question. Odds of winning a 3 dice vs. 2 dice battle are about 50/50. The invading army gets the advantage of an extra die, but ties go to the defender. It turns out that these advantages roughly balance each other out. (Full results.)

Question #2 is harder because battles between a large number of soldiers are complicated. It all comes down to who has to roll with a reduced number of dice. For example, if a large army is cut down early with a string of bad luck, its odds of winning go down much faster.

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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.

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