Board Games are Toally Math! Patchwork

Welcome to episode #2 or Board Games are Totally Math! The blog that uses mathematics and statistical analysis to pick apart great board game designs and give a glimpse into how they are made, and what decisions take them from good to great. The first addition was somewhat light on the numbers, so let’s fix that right now. Today, let’s unstitch Patchwork… (Get it, unstitch, patchwork, OK let’s begin)

Patchwork is a two-player game designed by Uwe Rosenberg. Take a look at the BGG recommendations forum, pick a thread at random that contains words like “two-player” or “couple game” and chances are someone has already suggested it. And with good reason, it is just that good. It’s a game where the casual gamers can compete with the more strategically minded without having to rely on the luck of dice rolls or card draws. Easily one of the best games in my collection, let’s take a look.

As always, I will assume familiarity with the game, so feel free to read the rules, watch a video guide or even find a copy and play a few games so you can follow along with us.

Patchwork is a lightly strategic game for two players about claiming patches of material to stitch together and make the best patchwork quilt. Hardly a theme to make the average board game geek salivate but this game delivers on gameplay so strongly that most shouldn’t mind. At its core, Patchwork is a game about spatial reasoning, fitting pieces on your board, finding the right shape to make a coherent whole. If you like Tetris (that’s just about everyone right) then it delivers on that same experience. But with a twist, whereas the Tetris God would often deem to screw you over with all the wrong pieces, that role falls to your opponent. There is always an engaging back and forth between the players of passive denial and forethought to get that crucial piece or deny your opponent what they need. The balance that Rosenburg achieved in this game is tremendous, as stated before, casual and strategically minded players can compete and most often not finish too far from each other. How does Uwe do it?

Balancing Act

One of the reasons I wanted to break down this game numerically is that it has puzzled me for a long time.  Generally games always finish with only a few pieces remaining and with no more than a dozen empty spaces on each player’s board at worst. Normally the winner is not apparent until the scores have been fully tallied, maybe one player had a fuller board, but not many buttons left, or vice a versa. And what’s more, a variety of tactics seem to be capable of winning the game.

To understand this balance, I have made a spreadsheet (which I will make available here) quantifying all the patch pieces in the game. I recorded their size in terms of squares, their cost in buttons and time, what income they provide and their overall footprint (a better explanation of the footprint to follow). Some interesting things happen when you start looking at some of the averages. For the purposes of this article I will work to one or two decimal places, full figures in the spreadsheet.

The end of a game of Patchwork is often for me, the most interesting and tactical part of the game. There is never that many pieces left at the end, meaning choice is restricted and players have to make do with and fight over what is left. Too many patch pieces in the game and these hard decisions would be lessened, too few and players would run out of pieces before they reached the end of the time track.

But that is easy to account for, there are two 9×9 boards to fill for a total of 162 spaces. The 33 patches have a total of 166 squares plus the 5 one square leather patches, players could in theory fill both boards and have 9 squares worth of patches to spare.

Well that assumes they are playing perfectly, quite unlikely. So where would they end up on the time track if this was achieved? Surely they would run out of time or end short playing like this. Well we can look to the averages to get an idea. The average size of a patch is a little over 5 squares. So a full 9×9 board would average about 16.1 patches, which is just perfect for both players since there are 33 patches in total. On average, a patch will send the player forward 3.2 places. Multiply that with the average number of patches to fill a board, with the full numbers you get 51.7 spaces moved. And there are 52 space on the time tracker. Coincidence? I think not.

What does this mean for our theoretical perfect average player? More than likely they are going to either limp over the finish line by a fair or fall just short. It gets even worse if we compensate for the leather patches (I considered them as taking up 2.5 spaces as an average, since most games players get either two or three leather patches). With the patches being accounted for, players should finish in around 50.1 time spaces, falling even further short of the finish line.

But that never happens, players never fall short of the end space, they never fill the board 100% either, which would only cause them to fall short of the finish line even more. Why then does it seem you can always make it to the final space of the time track?

On your turn, you can jump ahead of your opponent on the time track and claim buttons to the value of the amount of spaces moved. I used to think this was a bonus move, something to help you out, or a tactical way to end your turn and get some extra buttons. Now I am starting to think this is actually a penalty. It makes sense, it is after all, the only option you have if you cannot afford, or fit on your board, any available patches. We know from looking at the averages that you can in theory fill the board in time, but more than a few hops ahead of your opponent and this would be impossible. You get extra money for jumping ahead, but each time you do limits the total number of purchases you can make.

Yup. I think my mind is made up. Jumping ahead is a penalty. It is a penalty for spending too much or not leaving yourself the room to place patches. There are some tactical uses for it, say if you want your opponent to buy something and put you in the right spot to buy that piece you need, but overall, it is a mostly negative move, with a few benefits. Like most of the choices in this game, it is a….

This is something you may have intuited playing this game, but every choice you make has a lot of pro’s and con’s. It’s one of the things that makes this game so balanced. It is very hard to say what is the right and wrong decision at any given time, perhaps there is no wrong answer. There is good and bad to every choice, you just have to trust your judgement and do the best you can.

You can see it in nearly every patch you pick up.

This one costs nothing and has a middling time cost, but it is big and bulky and awkwardly shaped. Particularly in the late game, it can be hard to place anywhere useful.

This one has an income of three for only seven buttons, but it is small and has a big time cost.

I thought this game would lend itself well to analysing the patches statistically, but I think the balance is achieved through more than just raw numbers. There are some observations you can make. I decided to add together the button and time costs* of all the patches and if you group them into the patches with zero, one, two or three income. You see that they total costs do not deviate much within the groups.

*Since time in on the track can be exchanged for one button there would appear to be a 1:1 trade off in value between time spaces moved and the button currency, if you look at the three square patches they all add up to the same total, so I think it is safe to look at time and buttons as a total shared currency.

Gameplay Tip

It should go without saying that the patches that provide an income are best snapped up early, on average, they all take until you’ve passed three income buttons before they have paid for themselves. Didn’t need the numbers to know they are better the sooner you buy them, but towards the end of the game, particularly with three or fewer income buttons to pass, you maybe have to ask yourself if that three button patch is a great as it seems.

Rosenburg may have used some mathematics to calculate the costs for differing patches, but really there is more to each piece than their button and time costs and their income possibilities and size. You could possibly map those as some kind of three-dimensional graph, but I suspect it will not tell you a lot. The patches are also balanced on intangibles, like their overall footprint, or how awkward the shape is, or whether it is more than three squares long and wide so is useless once you get your 7×7 completed.

I took the averages of the three wide patches compared to the others and sure they cost a little more on average, but not much considering that they generally cover a bigger section of the player’s board and include three of the four patches with the maximum three button income.

There are just some elements of board games that cannot be designed and balanced entirely in a spread sheet. It is not a task that many designer will particularly enjoy, but I am certain Rosenburg went through a lot of design iteration and play testing to get the balance of the patches just right. I can only imagine how long that must have taken…

What’s on Offer?

Ever play the game where you start with a number (I used to start with 21), two players would take turns to subtract one, two or three from the number, whoever was able to subtract the last number to make the total zero was the winner. Kind of a silly game, one like noughts and crosses (tic-tac-toe) that is fun when you are a kid, but you soon grow smart enough to break the game and it becomes boring. Soon you realise that the way to win is to give you opponent the number 4, that way whatever they subtract, you can subtract the remainder and win. Then you realise that if you give them 8 then you can be guaranteed to be able to give them 4 on the next turn, then 12, 16, 20. Pretty soon you can win every game – until your opponent figures it out too, then the winner is whoever goes first. Game broken, what else can we play.

It’s a bad game, but Patchwork borrows from it in an interesting way. The way you buy patches and move to that spot is much like subtracting from the number, there are times when you can see the ideal patch to fill that awkward space on your board, it is possible to position the wooden pawn so that you can be sure to get it. That is unless your opponent is trying to do the same…

I do wonder if this is why the game set up starts the wooden pawn in the space clockwise to the one two square piece. It can be really handy for patching a gap, and it is often fought over when I play in the fashion of the subtraction game. The way that the selection of patches changes throughout the game just adds another layer of depth and fun. There is a great feeling when you can position the wooden pawn just right to mess over your opponent, skipping the piece they want badly, putting them in front of three patches they cannot afford or forcing them to set you up for the next patch you want.

The 7×7 Token

I have puzzled over the 7×7 bonus. I am not a fan of game mechanics that let the person in the lead get an even bigger lead. It would seem that way on the surface, but I think there is more to it than that. I don’t think the first to get a 7×7 block is necessarily “in the lead”. In fact getting the 7×7 block can be a hindrance, once you make the 7×7 square, you will have only a narrow border on two to four sides of your board, this can severely limit the patches available to you, pretty much all the three square wide patches will be impossible to place.

I think the token is an incentive for having a neat and orderly pattern rather than being the quickest, after all, the game is not a race, you both reach the end in the same number of spaces. On a side note, it is probably one of the few things that could be considered thematic, a patchwork quilt with irregular edges is better than a quilt full of holes. It is also another competitive element, there often comes a point in the game when both players realise how close they are to the 7×7 shape and suddenly you are in a battle to get there first.

I think it is an encouragement to play in a certain way, maybe compensation for when you cannot fit the larger pieces on your board but your opponent can still buy them.  It is however, one of the few aspects of the game’s design that I am dubious about. I would feel better about it if I could remember any time when it had actually helped the lower scoring player catch up. I feel like the winner of the game is usually the one with the 7×7 token, and usually they could have won without it. Makes sense, the neater your board the better you are likely to do, and with the patches bought and placed more densely  you will surely get more income from passing the buttons on the time track.

It seems as though, more often than not it just makes the winner of the game win by even more, which has the odd quality of being both useless, and feeling unfair to the loser. It may be interesting to play without the token to see how it affects the game. Or maybe just use is as a tie breaker.

This may be more like anecdotal evidence than objective observation, I really ought to get some final score data rather than going on my gut. If any of you readers keep track of your scores and who got the 7×7 token I would love to take a look. Or just tell me your experience with this feature.

Sewing It All Up

I hope this gave you an insight into the design of this game. I know it has helped me appreciate Rosenburg’s work on this game more. For me, looking at the game in detail, it is like seeing a beautiful work of art or watching a movie with a plot that just blows you away. I get that same feeling of jealous admiration, this game makes me think “damn you, how can you be so brilliant at game design.” Which is a great source of inspiration and motivation. But it also illustrates how simple the answer is, hard work and persistence. I guarantee you that the first 33 patches Rosenburg cut out of card were not the same ones you get in the box. Designing a great game means designing way more content then you need, throwing out what doesn’t work and refining the rest through playtesting and tweaking and repeating until you get something that delivers the experience you want.

Really the numbers are only half the story here. I suspect if you could quantify and map a graph of cost against reward for each patch it would fit a normal distribution. It’s what makes the puzzle of filling your board so engaging. There is no clear “right” solution, a patch that is great for you may be useless to your opponent. The game makes you adapt and rethink your tactics on every turn. And for a game that from the outside looks like multiplayer solitaire, there is a lot of player interaction.

Let me know your thoughts and what have you in the comments. Suggestions for games to study next are welcome. I’m thinking Ticket to Ride might be fun to look at. It makes me think back to A-Level maths and having to calculate shortest routes or fewest node hops. Won’t that be fun…