Puerto Rico and its blue and yellow socks

We had an excellent game day today, before the snow started falling in downtown Seattle. We opened with quick games of Dominion, switched to 5-player Race, and then we hit Puerto Rico with 5. (We wrapped up with 4-player Settlers and a final game of Race.)

They were all games played with good gamers (who are also good friends). There's always a bit about what feels random...it was on my mind after reading the "Statistically Speaking" article on Gamasutra (link in the sidebar). The topic crept into tonight's game, as we started Puerto Rico. How do you determine who goes first? My standard way is take 5 barrels, 4 corn, 1 indigo...whoever gets the indigo starts. Fast, quick, doesn't need dice or prolonged discussions of how to best use rock-paper-scissors in a group environment. (I'm also of the belief that RPS is not random, and would not be useful in determining a necessarily random behavior!)

My friend said the magic words of "Wait - let me do it!" He has a new way. Take 3 indigo tiles, 2 corn tiles, and let players take tiles clockwise until the system is revealed. For example, Player A takes a tile, reveals indigo. Player B takes a tile, reveals corn. This would mean that Player A is third player with the last indigo, Player B is fourth player with the first corn! (In Puerto Rico, the first three players in the order start with indigo, the last 2 players get corn.)

That certainly seems magical, right? Something seemed fishy, but the bottle of wine that we had been sharing in celebration of the snow seemed to lighten our spirits...so we continued as it was decided. (He showed the system for Indigo/Indigo/Corn or Indigo/Indigo/Indigo - you only needed to go as far as three players.)

Of course, being gamers we yearn to have complete knowledge. We argue over what the best Settlers placement was, whether it's better or worse for random distribution of numbers, ways to eliminate/reduce/create 'luck' in favorite games...and we will now circle back to the magical method.

It makes for a great scam.

We'll start at a table where I'm playing with Alice, Bob, Charlie, myself, and my buddy Jack. I'm the person getting everyone to sit down, and I will use the magical method as described above. I'll choose Alice to start, going clockwise around to Bob and then Charlie as needed. 5 face-down tiles, 3 indigo, 2 corn - just like the game.

Yes, it's true that any player has a 60% of starting the game with an indigo, or a 40% chance to start with a corn plantation. But the next step is where it breaks - we're not dealing with replacement...they're related events.

Let's start with that 60% indigo tile draw for Alice. Alice is happy - she is now one of three indigo players, and therefore has a 1 in 3 chance of going first! Not quite. Bob has 4 tiles left - 2 indigo, 2 corn. If Bob gets a corn tile, he is first corn and therefore Alice is third indigo. Bob draws corn half the time - half of that 60% we had, Alice goes third (30%). If Bob gets indigo, Charlie has 1 indigo, 2 corn left. If Charlie gets a corn (a 2 in 3 chance), he becomes first corn, and Alice's indigo draw earlier puts her as 2nd player. (20% chance, as it's 2/3 of 30%.) The remaining 10% chance is Alice's chance to go first.

Alice could have drawn corn, 40% of the time. Alice would be happy again, assuming she hasn't learned her lesson from indigo discussions. There's a 50/50 chance that she is first corn player! Sadly, the magical method will fail her here as well. Bob would be left with 1 corn, and 3 indigo to draw from - 3 times out of 4, he'll draw an indigo, making her last player.

Alice's breakdown:
1st player: 10%
2nd player: 20%
3rd player: 30%
4th player: 10%
5th player: 30%

Now, if I were a devilishly clever person of low morals, and wanting to be first indigo or first corn (first player & fourth player, respectively), I would look to the person to my right and say "Hey, choose a tile and reveal it for yourself. Now do the same for me, as I finish setting up the game!" 60% of the time, I'd start as first corn or first indigo.

And that ends the story of the horribly magical method, that you should never ever use in polite company...or mine.