“Terraforming Mars” – and why I think a board game like this is worth your time

Hello good people!

Recently I acquired a new board game dubbed Terraforming Mars. The game is developed by Jacob Fryxelius and published by Stronghold Games. It has a price tag of around 60 EUR / 45 USD on Amazon. The game has an impressive rating of 8.4 on BoardGameGeek.com.
The game can be played by 2 to 5 people. The box says a game should take 90 to 120 minutes, but in my experience it’s more like 2 to 3 hours (I’ve played 20+ games so far).

The objective and how the game is played
You and your fellow gamers will be assuming the role of big corporations working to … *queue suspense sound effects* … Terraform Mars *fanfare*. Not that surprising really, is it 😉
You start out by having a set production for your corporation, some starting capital and a corporate feat, which may help you throughout the game to a greater or lesser extent.
The games is played in rounds (simulating generations in timespan), where you go through phases of producing resources, buying project options, spending money to built said project or from a set of standard projects. Players take turns performing one or two actions per turn. When everyone is done, the round ends and a new one begins. Each project can have a variety of effects, of which some raises one or more of the three global parameters (oxygen, water and tempetature). When all these parameters reach a set level each, the game ends.
During the course of the game you raise your Terraforming Rating, which ultimately serves as Victory Points. The player with the highest amount of victory points is the winner.
The game is composed of both cards and a board on which you play tiles, thus giving an additional way to interact with other players.
You can find the rules here; Terraforming Mars rules

Why I love it and what I think makes it great
Terraforming Mars has me hooked on a number of things, which is why I’ve played more than 20 games in the course of 4 weeks. First off it has a set of really interesting mechanics. I’ve found myself pondering over which projects and actions are the best, what the return on investment is on each, how I should sequence them and lastly what my opponents are doing. I’ve played 2-player games by far the most. Even here the actions taken by my opponent in these games are an important enough factor, that I could never successfully disregard this.

Another interesting thing is that the resources you produce over time enable you to complete more projects, making you produce even more resources over time. Thus building up resource production is the most important thing in the beginning of the game. But when the game end all these resources are worth nothing in terms of winning the game – only Victory Points is – which brings in a whole new aspects of when should you shift your focus from producing resources, to gaining Victory Points.

The setting of the game is great too. It’s really well executed with regards to mechanics and gives a great ambiance to the game as a whole. I’ve caught myself chuckling a few times, when I read the flavor text on a project card and seen how well the story on it supports what it actually does in terms of gameplay.

Complexity wise the game has a lot to offer. More than once I’ve found myself missing things, so if I had to say something negative it’s got to be that the game is best played when you’re not too tired, as it will affect your enjoyment, as misplays will happen more frequently.

Terraforming Mars does have an element of luck, but I’ve never felt that you couldn’t make skill count too. It balances a fine line between skill and chance, and it does it well.

Ending remarks
I see myself playing this game over and over, and are trying to find pockets of time during my week constantly.
Terraforming Mars has all the good elements that you could stuff in a box and I am happy to toss countless hours into managing my interstellar corporation for generations to come.

Kudos FryxGames.se on a job well done 😀

10,000 version 1.6 is up on Google Play

Hi,

The 10,000 app has recieved an update.
I’ve made some improvements to the User Interface, so that buttons will be easier to tell apart from the rest of the UI. There has also been made some modifications to the computer player, so that it will try and score more than you, if you’ve reached 10,000.

I hope you all enjoy it. Go check it out 🙂

Best Regards
Lars Sonne

The Android likes dice – 10,000 version 1.5 released

Hi all,

I’ve released a new version of the 10,000 – The Dice Game app on Google Play. This release features a few things.

  • You can now play against the computer
  • German translation
  • Better randomness of the dice rolls

The option to play against the computer is something that has been on my mind for quite some time and I’m happy to say that the result is now here. The AI proved to be a bit more challenging to implement than I first thought, but from my time in the computer industry I was hardly surprised. Things are in my opinion never straight forward 😉 Nevertheless I’m quite pleased with the result.
I’ve aimed for the computer to be beatable but not a walkover, relying on a set of simple rules to govern its behaviour. I also tried to make it animated so you can follow the steps it takes when playing and who knows, maybe you’ll even get a new trick or two. I hope you’ll enjoy it!

Happy rolling 🙂

10,000 version 1.4 on Google Play

Hi all,

I’ve put a new version of “10,000 – The Dice Game” (there’s a link at the bottom of the post) with a few new things in it.

  • Bug Fixed! One of the user, Brian Farmer, took the time to write a review, where he described an error where the game was reset if your screen went to sleep. That bug is now fixed and your game will now live on, while your phone takes a break. Thank you, Brian.
  • Cookie Policy. A popup appears when you start the app for the first time (or the first time after you update to 1.4). This warns the user about the use of cookies and provides a link where they can read more about it. This is a requirement to comply with the European legislation and the Google User Consent Policy. I chose to not restrict the message to EU citizens only.
  • Danish Translation. I am Danish after all, so why not 😉

 

As the astute reader may have noticed, the Google Play listing has also been changed a bit. A user, Daniel Der, made me aware the a comma is the correct digit grouping sign. That has also been corrected in the game too. Thank you, Daniel.

 

That is it for now. I’ll be eagerly awaiting the data on how fast the updates are getting to the users and maybe even make a post about that.

 

Best regards

Lars Sonne

 

And there’s the link I promised 🙂

Building an AI (part 2)

This is part 2 of a series on building an AI for the dice game “10.000”. To read part 1 go here.

 

The mission

My goal is that at the end of this post I’ll have a set of formal rules that can be coded into the AI. I want to know what the odds are for scoring when rolling any number of dice and how many points I can expect to score. I also want to make a simple decision tree for when to roll and when not to roll.

 

Calculating the odds

Since I’m no genius when it comes to statistics, I read a few articles on the dice probabilities at different rolls. For those of you that are interested you can find some mathematics explained here, here and here. These articles give us a basis for determining what the chances are of not being able to score points on a given roll.

In the following I’ll be listing tables showing the number of combinations that score points and the probability of these when rolling X dice (the columns “# successes” and “Success %” respectively). Conversely I’ve also listed the chance of not rolling points (the “Failure %” column). In the first table I’ll assume no 3-of-a-kind has been rolled or what I’d like to call basic probability.

The total number of combinations is calculated as 6^x, where x is the number of dice rolled.

Basic probability

Dice # Successes Success % Failure %
6 45.576 97,69 % 2,31 %
5 7.176 92,28 % 7,72 %
4 1.092 84,26 % 15,74 %
3 156 72,22 % 27,78 %
2 20 55,56 % 44,44 %
1 2 33,33 % 66,67 %

 

When rolling 1, 2 or 3 dice it also becomes relevant if there has been rolled 3-of-a-kind previously, since it increases your odds of rolling dice that can score points. In the second table I’ll assume that there has been rolled 3-of-a-kind of 2’s, 3’s, 4’s or 6’s, since these affect the odds.

3-of-a-kind already rolled

Dice # Successes Success % Failure %
3 192 88,89 % 11,11 %
2 27 75,00 % 25,00 %
1 3 50,00 % 50,00 %

 

Diving deeper into the combinations

These odds can be divided further into the different combinations of dice. I’ll list a separate table for every number of dice rolled, sorted by their success probability in ascending order. The column “Combinations” is the name of the combination you scored. “# Successes” and “Success %” express the same as before.

The row “Two 3-of-a-kind” have been included since we use all the dice and unlock all 6 on successive rolls. The row “1 or 5” is for the cases when none of the other combinations occurs.

 

Rolling 6 dice

Combinations # Successes Success %
6-of-a-kind 6 0,01 %
5-of-a-kind 180 0,39 %
Two 3-of-a-kind 300  0,64 %
Straight 720  1,54 %
Three pairs 1.800  3,86 %
4-of-a-kind 2.250  4,82 %
3-of-a-kind 14.400  30,86 %
1 or 5 25.920  55,56 %

 

If you sum up “# Successes” you’ll luckily end up with 45.576, which is the same number as listed in the basic probability table above. In the next two tables the process is repeated for four or five dice.

 

Rolling 5 dice

Combinations # Successes Success %
5-of-a-kind 6 0,08 %
4-of-a-kind 150 1,93 %
3-of-a-kind 1.500 19,29 %
1 or 5 5.520 70,99 %

 

Rolling 4 dice

Combinations # Successes Success %
4-of-a-kind 6 0,46 %
3-of-a-kind 120 9,26 %
1 or 5 966 74,54 %

 

The next three tables are for rolling one, two or three dice without having rolled a 3-of-a-kind previously.

 

Rolling 3 dice

Combinations # Successes Success %
3-of-a-kind 6 2,78 %
1 or 5 150 69,44 %

 

Rolling 2 dice

Combinations # Successes Success %
1 or 5 20 55,56 %

 

Rolling 1 dice

Combinations # Successes Success %
1 or 5 2 33,33 %

 

You could also be rolling one, two or three dice when you’ve already rolled 3-of-a-kind. The following tables shows, where I’ve added the row “Add to 3-of-a-kind” for this. Remember the odds only change when the 3-of-a-kind is 2’s, 3’s 4’s or 6’s.

 

Rolling 3 dice with 3-of-a-kind already rolled

Combinations # Successes Success %
3-of-a-kind 5 2,31 %
Add to 3-of-a-kind 91 42,13 %
1 or 5 96 44,44 %

 

Rolling 2 dice with 3-of-a-kind already rolled

Combinations # Successes Success %
Add to 3-of-a-kind 11 30,56 %
1 or 5 16 44,44 %

 

Rolling 1 dice with 3-of-a-kind already rolled

Combinations # Successes Success %
Add to 3-of-a-kind 1 16,67 %
1 or 5 2 33,33 %

 

This covers the odds for every combination of dice.

 

Weighting the odds

We now have some probabilities down that we can use to guide our decisions (and the AI’s) on whether or not we want to roll the dice at any given point. However if you just decided to roll based on the probability of scoring alone, you would not be playing optimally i.e. scoring the most points.

Enter Expected Value and Opportunity Cost.

So what is this?

Expected Value (or EV) is the value we statistically can anticipate getting over the long run from our roll. In this case the score we can add on average. The Opportunity Cost is the value we give up by rolling; being the points we could have banked instead of rolling.

How do we use that when deciding whether or not to roll?

What we basically want to be doing is weighting out the two, by saying if my opportunity cost greater than my EV, I shouldn’t roll. But why is this correct? Let’s take an example. You have one dice left and have already scored a 1, a 5 and 3-of-a-kind of 2’s. Your chance of rolling and scoring is 50%. You could either be rolling a 1 scoring 100, a 5 scoring 50 or a 2 scoring 200 more. This would give you an average score of 116,67, but only on 50% of your rolls, giving you an EV of 58,33. Your opportunity cost is your current unbanked points of 350 times the chance you’ll miss out on points on the next roll giving you 116,67. Since the opportunity cost is greater than the EV you should be banking instead of rolling here.

To put it simply:

If EV > Opportunity cost, then roll.

Otherwise bank the points.

This is not the complete answer though. It would be if we always banked after this roll, but we could choose to roll again and again. Therefore we need to factor in the EV of successive rolls, at least to some extent. But how do we do this, since you can score and therefore remove a different number of dice after each roll? I’m certain that smarter and more mathematically experienced people would be able to solve this puzzle, but I accepted that the heuristic I’m building will not guarantee that the optimal solution would be found every time, which I why I’ll believe I could employ a short cut here. Calculate the chance of having scored all six dice at the end of the roll when rolling any number of remaining dice. Then weight this by multiplying by the EV of rolling all six dice. This post is however getting quite lengthy, so I’ll save this part for now.

 

The Expected Value of a roll

Working on the combinations I listed earlier I will now map out the EV of every combination on every number of dice rolled. By doing this we get the last set of numbers we need to make an equation for when to roll and when to not. The assumption is that you’ll score the maximum number of dice.

 

Rolling 6 dice

Combinations EV
6-of-a-kind 2.000
5-of-a-kind 1.525
Two 3-of-a-kind 1.000
Straight 1.000
Three pairs 750
4-of-a-kind 1.050
3-of-a-kind 575
1 or 5 156

 

Rolling 5 dice

Combinations EV
5-of-a-kind 1.500
4-of-a-kind 1.025
3-of-a-kind 550
1 or 5 139

 

Rolling 4 dice

Combinations EV
4-of-a-kind 1.000
3-of-a-kind 525
1 or 5 121

 

Rolling 3 dice

Combinations EV
3-of-a-kind 500
1 or 5 105

 

Rolling 2 dice

Combinations EV
1 or 5 90

 

Rolling 1 dice

Combinations EV
1 or 5 75

 

Rolling 3 dice with 3-of-a-kind already rolled

Combinations EV
3-of-a-kind 500
Add to 3-of-a-kind 3-of-a-kind value * 108/91
1 or 5 105

 

The fraction in the “Add to 3-of-a-kind” is to account for doubles and triples. I counted these by hand.

 

Rolling 2 dice with 3-of-a-kind already rolled

Combinations EV
Add to 3-of-a-kind 3-of-a-kind value * 12/11
1 or 5 94

 

Rolling 1 dice with 3-of-a-kind already rolled

Combinations EV
Add to 3-of-a-kind 3-of-a-kind value
1 or 5 75

 

That should be enough numbers to feed into the equation.

 

What if any kind of special rules take effect when playing the last round?

During the last round we could use the equation, but it is not enough to bank any number of points. It only matters if your total score exceeds that of all of your opponents. A simple solution is to keep rolling until you have more points (both banked and unbanked) than your opponent.

Put simply:

If banked + unbanked points <= opponents score, then roll.

Otherwise bank

 

That should be enough to implement a set of rules for the AI.