# How Sagrada’s Difficulty Scales with Players

*Sagrada* is an interesting game. Based around the idea of building stained glass windows, it plays more like a puzzle than it does necessarily a competitive board game. Each turn dice are rolled, and chosen to fill in a partially set pattern on each player’s board. If you haven’t played *Sagrada *then you can read the review we wrote about it in 2018 here.

*Sagrada *is a beautifully put together game, and being a game about dice it is also a board game analyst’s dream. There are so many things that can be taken and pulled apart, from specific goals to dice counting to the very concept of the game. Today though, we are going to fixate on an interesting thought I had when playing a two player game recently, and pull it apart like a bear in a pulled pork factory. Here is the thought –

Difficulty in Sagrada is inversely proportional to the number of players in the game.

I told you it was interesting.

## Difficulty in Sagrada

Okay, so before we continue there are a few things to take into account. *Sagrada *has a fairly intuitive difficulty system, wherein players can select at the start the window they want to try and build from a selection of four (or two two-sided) window designs. Depending on the difficulty depends on the number of tokens that player gets to spend on tools throughout the game. For this we are going to assume that the same difficulty, say a difficulty level of 4.

Now, already, there is something worth considering and that is the math behind *Sagrada*. *Sagrada *is a mathematician’s dream, and one of the many reasons behind that is the basis of the game has its core in algebra.

Each round, as players, you draw:

2n+1

dice from the bag, where* n* represents the number of players. This is crucial to our thought process that difficulty is inversely proportional to the number of players, since all players have 20 spaces on their board to fill up. Some of those will have colour criteria, and some will have a number. Some will have no criteria at all. That’s just how the game is.

On top of that there are the orthogonal placement rules – namely, no two colours and no two numbers can be next to one another. This means that actually, whilst you may think placements are open for you, there are limiting factors – and a lot of them – making *Sagrada *a beautiful puzzle.

That being said, what this also means is that, due to the algebra, the difficulty scales…but scales backwards.

## How Sagrada Difficulty Scales.

So, before we begin, we have a solid number of spaces that need to be filled up on the board – 20. That is a solid and unchanging, rational number and one that is made possible through 10 rounds, in which the players draw two dice from a pool of 2n+1.

The order the round goes in is (where P is the player) P1, P2, P3, P4, P4, P3, P2, P1 in a four player game.

What this means, in a four player game, is P1 has 2n+1 dice to choose from. P2 then has (2n+1)-1 dice to choose from, since one dice was chosen by P1. This can be shown as (2n+1)-p, where lower-case “p” represents the number of players who have already gone.

This means that in a 4 player game, P1 has 9 dice to choose from, and P2 has 8 to choose from. P3 has 7, P4 has 6 and P4 has 5. P3 then has 4, P2 has 3, and P1 has 2. The left over dice goes to the round tracker.

As you can see, the graph of this is really boring to look at.

Each and every turn your own player board will be filled up by two dice (in theory) which mean that every turn you board is progressing to a state of 20-(2 x the round number), or 20-2r.

Why does that matter? It doesn’t, I’m just having fun with algebra.

Okay, so silliness aside, that 2n-1 is incredibly important. If you are Player 1 in a 4 player game then you will have to choose 1 in 9 followed by 1 in 2. If you are Player 2, you will get 1 in 8, followed by 1 in 3. If you are Player 3, you will get 1 in 7, followed by 1 in 4. Finally, Player 4 will get 1 in 6 followed by 1 in 5.

If you are in a two player game, however, things change. Instead of drafting 9 dice to begin with you only get 5. This means that you still have the same 20 spaces; however, as P1 you only get 1 in 5 to choose from, then P2 gets 1 in 4 and 1 in 3. P1 then only gets 1 in 2 again.

This means the dice have to be more in your favour in a two player game as you need to roll more precise numbers.

## Turning It Around With The Law of Averages

Okay, so we can simplify by looking at averages. There are five coloured dice in *Sagrada* – blue, red, purple, yellow, and green. There are six numbers on each dice.

When you draw 9 dice, the odds of any one colour on any one dice is 1/5 (the same no matter how many dice there are – those are the odds); however, the odds are there will be 1.8 of each dice colour on the table. Needless to say, you can’t have 0.8 of a dice, but bear with.

When you only have 5 dice, when you are only drawing 5, there is a 1/5 chance of drawing each colour – thus the law of averages suggests that over the course of the game there will be 1 of each colour on the table each round.

That means we’re looking at a 1.8 on the table vs 1. Likewise, we can do the same mathematics with numbers on dice, the odds are, with 9 dice, there will be 1.5 of each number on the dice each turn. With only 5 dice, the odds are way worse at 0.83 dice per hand having the number you want when you are after specific numbers.

### …Making…More Sense…

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.

This means the odds multiply. We’ve covered this before in the luck mitigation article a couple of weeks ago, but essentially, we look at the odds of failure. Roll 1 dice and, with the colours, there is an 80% chance of not getting what you want. Roll 9 dice, and the odds are there is a 13.42% chance of not getting what you want, or a 86.58% chance of getting what you want.

Roll 5, and there is only a 67.23% chance of getting what you want – when it comes down to colour.

With numbers on dice, there is a 81.31% chance of getting what you want via 9 dice, but only a 60.61% chance on 5.

Naturally, this scales depending on player count, and it varies on position in the playing order you are, but the maths are sound. You stand a higher chance getting what you want, when you want a specific number or colour, when more dice are rolled.

Seems obvious when put like that…

## Difficulty in Sagrada **CAN** be Inversely Proportional to Player Count

**CAN**

So, there we have it – a bit of mathematics today to prove that difficulty can be inversely proportional to player count. Of course, this is all theory. In reality there are too many variables – everything from the luck of the draw to the actual windows chosen can affect the game.

So, what are your thoughts? Do you believe the numbers are accurate? What’s your experience of the game and player count? Let me know in the comments below.

Enjoyed this, Luke! I can’t believe you did your original review in 2018! π

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Haha I know, right? It was actually one of the first games we bought at the UKGE 2018, so it would have been mid-2018 as well. Speaking of which, where has 2019 gone?!?!?

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