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Puerto Rico is generally considered as a game of mostly skill, with very little luck involved. In this regard, it is more like chess or checkers, than the typical board game that uses cards or dice.

The only random factor I know of in Puerto Rico is the fact that the plantation tiles are initially turned face down, and shuffled, then a number of them are turned face up and made available, with redraws from the face down pile as the game progresses. In theory, this could lead to one of each plantation being made available at any given time, but it could also lead to a (random) abundance of some, and an absence of other choices.

What (if any) other random or luck features exist in the game?

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The first random thing that happens in Puerto Rico is turn order. This can be offset by having players bid on seats.

If you are playing with the expansion, then the buildings available are randomized. This can also help to prevent the game from being solvable.

There are the plantation tiles as described int he question.

Finally if you are playing with more than 1 other person, player actions are in a way a randomization. Some moves become better if your opponents act perfectly logically, others are better if your opponents make mistakes, and you can't know what they will do. This is unlike chess (or a 2 player Puerto Rico game) where your opponent making a mistake cannot be a determent to your game.

Overall Puerto Rico is a very low luck game with all randomization being "input randomization". This means that all random factors are chosen early in player decision, and by the time a player acts, they will know the result of at least that action.

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There's also randomness if you have the Hacienda. –  seppo0010 Oct 22 '13 at 18:30
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If you are playing with the expansion, then the building are drafted. –  The Chaz 2.0 Oct 22 '13 at 21:50

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