Okay, dear reader, let’s follow a thought to its ultimate conclusion. In the last analysis article we looked at how Sagrada difficulty scale varies with player count. It turned out some pretty interesting numbers, giving the concept that the odds change based on the player count due to the number of dice.

Then, towards the end of the article, came this idea –

To simplify – if you need a specific number you have a 1/6 chance of getting it per dice. If you want a specific colour, you have a 1/5 chance of getting it per dice. If you want…just…anything, then it depends on what it is orthogonally adjacent to. I’ve just realised there is a much bigger article in here, but let’s continue on the train of thought for now.

So, today we are going to explore that thought. Although I try not to posts runs of articles on the same game (leaving aside my recent D&D series for new players) this one is too interesting to let go – here is the thought –

Depending on where you are in the player order varies your odds of winning a 3 or 4 player game… based on the probability of something specific (number or colour) you want coming out each round. Since play proceeds clockwise, we can then work out over 10 rounds, which starting position has the best odds of winning the game from turn one.

Hacking Sagrada – The Assumptions

Okay, so we have to make a few assumptions. Firstly, one from my side – I am not going to explain how the game is played here, so will assume you know the game. I will assume you understand the rules and the layout.

In regards to working out the math – we are going to make the assumption that you are only after filling one specific space at a time. Where this is never really the case in the early game, as the game progresses it plays more of a role, where you need a specific number, specific colour, or specific combination to make something work.

Is it sad that I’m getting really excited about this? LET’S DO THIS.

Sagrada – You may have seen this picture in the last article…

The Numbers Behind Hacking Sagrada

There are a series of facts we need to state to fully have all of our pieces on the board. We’re going to lay them out there and then we can play with them as and when we need to.

1. There are 20 squares each player needs to fill.
2. There are 10 rounds.
3. There are 5 colours of dice.
4. There are equal numbers of those 5 colours in the bag of dice – with 90 dice in the bag, meaning there are 18 of each.
5. There are 6 numbers on each dice – 1 through 6.
6. The odds of drawing any one colour is 1/5. The odds of rolling any one number is 1/6.
7. Sagrada can be played with 2, 3, or 4 players.

A Note On Player Count

Now this here, this very moment, is the moment it gets really interesting. In a 2 player game, each player will go first 5 times. This is because it cycles around them.

In a 3 player game, well 10 rounds doesn’t divide by 3 players. This means all 3 players will go first 3 times, and 1 player will go first 4 times.

In a 4 player game, 2 players go first 2 times, and 2 players go first 3. This is the crux of the debate. If your odds of getting the dice you want depends on when you get to pick them, when is the best time to go in the overall playing order per game?

Can you feel the excitement? I can. My math bones are tingling.

QUICK ROBIN – TO THE BAT EXCEL!

Working Out The Odds in a 4 Player Game

So, what we are looking at here is the odds per player per turn of having a specific colour you want at any turn during the game. Please note, at this moment in time we’re not looking at numbers, but rather, judging the specific number of dice you roll in a 4 player game (9 per turn) what the odds are of there being a colour you want on your turn based on all other factors being equal.

What this table does is take both turns into account, multiplying the odds of getting a colour you want in both rounds. So, Player A has picks 1 and picks 8 to take into account. For Player B, we have picks 2 and picks 7 to take into account. You get the idea. This is because the choice order per round goes Player A, Player B, Player C, Player D, Player D, Player C, Player B and then Player A again.

So, in Round 1, Player A has a fantastic chance of getting what they want on their first pick, but really poor odds on their second pick. As we can see, this balances out to a pretty rotten score.

We can extrapolate that across all the rounds, and then we start seeing something really interesting. So, in Round 2, Player B will go first. In Round 3, Player C will go first. In Round 4, Player D will go first. You get the idea – this rotates for 10 rounds.

What is interesting here, is that after 8 rounds each player has been first twice, meaning the game balances out. The game is not 8 rounds however, and so we end up with a slight skew. If we average out the odds across the game…we actually get this –

Yes – the odds of getting precisely the colour you want every time you pick, is actually more if you are the last player at the start of the game. It’s really interesting, right?

We can work out similar odds with the numbers on the dice. The maths are slightly different, as we are working out odds based in 1/6 possible options rather that 1/5, but the pattern is the same.

As you can see – the pattern is actually identical.

So, in a 4 player game, if you are after a specific number of colour, you are best to go last. Pretty interesting, but what about 3 player?

Hacking Sagrada in a 3 Player Game

In a 3 player game of Sagrada there are a few things that change. Firstly, one fewer player means one fewer player going first more than the others. Since numbers and colours of dice give the exact same pattern in regards to odds then we are only going to look at one of them for working out the odds for a 3 player game. Who are they highest with?

Well, the mathematics are worked out the same way. We’re looking at probability, across 10 rounds of the game. This time it goes Player A, Player B, Player C, Player C, Player B, Player A. Once again, we will work out the odds of having exactly what they want in both picks of a turn.

In a 3 player game there are only 7 dice, so this needs to be taken into account as well.

The below table shows what the odds look like across 10 rounds (we’ll work out the average after). As you can see, with fewer dice the odds of getting what you want are actually more extreme due to slimmer pickings, going from just under 30% to almost 40% chance depending on your position in the playing order.

What this means is that very final round really swings the odds. This is because until that moment everyone has had 3x turns as the first player. This swings the average – and again we are looking at a graph where the odds shift slightly in favour of going last.

Again, we could recreate these graphs for going after specific numbers as well, but the pattern is the same.

So, generally speaking, being the last player in the first round of Sagrada should, in theory, put you in the best stead. I told you it was interesting.

…Buuuttttt…That Isn’t How Probability Works…

Okay, so here we have to put our hands up and point out one of the many holes in this analysis. The swings in odds are less than 3% (percentage points) across the graphs. There is actually very little difference between the numbers, but the graphs are truncated to display the difference better (blame Excel, it does it automatically). What this means is, yes, there is a swing to being last – statistically speaking having the best choice available depending on what you are after; however, probability, by default, is about predicting (and in board games often mitigating) randomness.

What all this means that you could play twenty games in a row where other players steal what you want before you, or a hundred games where the dice just won’t roll right for you. All working out the odds like this means is that in theory, and purely in theory, going last gives you the best choice. If you played 200 games in controlled conditions, with equally skilled and caffeinated players, then the odds are there could be a slight swing towards the players who go last during the first round of the game winning.

In theory…or not…that’s just how the universe works.

Hacking Sagrada – Conclusion

So, assuming everything plays how it should, going last in theory gives the best odds of winning a game of Sagrada if all external influences are the same. This means we aren’t taking player skill, environment, potential dice bias, or anything like that into account. It’s an interesting theory though.

Please feel free to leave your thoughts and poke any holes in the theory in the comments below.

RIGHT NOW ROBIN – ON TO THE NEXT BAT BOARD GAME!

Published by Luke

Board game nerd, comic book geek, Rubik's Cube solver, Splendor player, and Blogger. View all posts by Luke