The Math of Tak

Those of you who know me in real life know that I am a gamer. I’ve been playing board games of various sorts since I was a kid; Dungeons & Dragons and its ilk since high school; and LARPing (boffer and not) since college. Yes, I’ve somehow become even MORE nerdy as I’ve aged. Chalk it up to the friends I’ve made through gaming. Gamers, as we know, tend to attract more gamers. We flock together, able to talk to each other about our shared passion. And through that conversation, we discover new games and new ways to connect with our fellows. One of the side effects of this is a near-continuous (at least for me) hemorrhaging of money into the hobby as we discover new things. Over the years I’ve backed more than my fair share of Kickstarter campaigns for new games from various designers and companies, ranging from massive new RPG products like Numenera and Shadows of Esteren to simple things like Pairs. So, of course, when I found out about Cheapass GamesKickstarter for Tak, based on the books of Patrick Rothfuss’s Kingkiller Chronicle, I had to go take a look.

shut up and take my money

Gaming is why I don’t have any money.

Holy crap, this game is beautiful. The core idea is incredibly simple: You’re trying to make a path from one side of the board to the other by placing pieces of your color, while trying to prevent your opponent from doing the same. To do this, you can either place a piece or move one of your pieces (or stacks of pieces). That’s it. This video shows you the rest of the rules. It’s simple, elegant, and easy to learn. Then you start to play it, and you realize just how complex it can become, and how quickly things can turn around in the game. You can play it on any size board, from 3×3 on up. The beta rules provided as part of the Kickstarter go up to an 8×8, with some uncertainty as to the piece counts for a 7×7 board (saying that it’s an uncommon size).

Unacceptable


WHAT DO YOU MEAN, “IT ONLY GOES TO 8?!?!?!”

Being that I’m an engineer, and that I get paid to do math, I figured I’d do a little math to see what I could figure out about Tak. Because yes, I do math at things for fun sometimes.

Let’s start with the basics. I put together a table of data based on the beta rules available on Cheapass Games’ website:

Side Length Spaces Pieces per Player Capstones per Player Total Pieces Total Capstones
3 9 10 0 20 0
4 16 15 0 30 0
5 25 21 1 42 2
6 36 30 1 60 2
7 49 40 2* 80 4*
8 64 50 2 100 4

*Note that I’m going to just assume for ease of calculations that when you’re playing on a 7×7 board you get 2 capstones. It just makes sense to me with the pattern of the previous sizes: Every 2 extra spaces on a side gets an additional capstone. 3-4 gets zero, 5-6 gets one, 7-8 gets two, etc. Plus, as you’ll see soon, the math works out so beautifully.

From that table of basic information, I pulled out a few pieces of derived data to further expand what I could work with mathematically:

Side Length Total Pieces – Spaces Total Pieces/Spaces Spaces/Capstone
3 11 2.22
4 14 1.88
5 17 1.68 25
6 24 1.67 36
7 31 1.63 24.5
8 36 1.56 32

“Okay, Josh,” you say, “that’s neat and all, but what does it mean?” It means, dear reader, that we can figure out how to play on any size board we want. First, let’s take an actual look at these data. They make much more sense in graphical form:

tak stats 1

Okay, maybe they don’t…

So, what exactly are we looking at? The short answer is that it’s a mix of useful and superfluous information that, when examined under-*ducks thrown vegetables* All right, all right! We can use numerical analyses to figure out the slopes of the lines representing Pieces Per Player, Total Pieces, Total Pieces/Spaces, and Total Pieces – Spaces. By doing this, and then projecting the curves out to higher numbers of spaces per side, we can get a good idea of how many pieces would be needed to play games on those size boards. I say “a good idea” because, well, this is what the projections look like:

Side Length Spaces Pieces per Player
9 81 59
10 100 71
11 121 83
12 144 96
13 169 109
14 196 124
15 225 139
16 256 154
17 289 171
18 324 188
19 361 205
20 400 224
21 441 243
22 484 262
23 529 282
24 576 303

Kind of ugly numbers for the piece counts, right? The space counts are what they are, but we can round the piece counts to look a little bit nicer. We can also project out the number of capstones, using the formula I described earlier (+1 capstone for every 2 added spaces on a side). This gives the following table:

Side Length
Rounded Pieces per Player
Capstones per Player
Total Pieces
Total Capstones
Total Pieces/ Spaces
Spaces/ Capstone
3 10 0 20 0 2.22
4 15 0 30 0 1.88
5 21 1 42 2 1.68 25
6 30 1 60 2 1.67 36
7 40 2 80 4 1.63 24.5
8 50 2 100 4 1.56 32
9 60 3 120 6 1.481 27
10 70 3 140 6 1.4 33.33
11 85 4 170 8 1.41 30.25
12 95 4 190 8 1.32 36
13 110 5 220 10 1.302 33.8
14 125 5 250 10 1.28 39.2
15 140 6 280 12 1.24 37.5
16 155 6 310 12 1.21 42.67
17 170 7 340 14 1.176 41.29
18 190 7 380 14 1.173 46.29
19 205 8 410 16 1.136 45.125
20 225 8 450 16 1.125 50
21 245 9 490 18 1.11 49
22 260 9 520 18 1.074 53.78
23 280 10 560 20 1.059 52.9
24 300 10 600 20 1.042 57.6

What a god-awful mess! Visually, it absolutely is (I’m an engineer, not a graphic designer!). Mathematically, it’s quite beautiful. All of these things are described by simple, well-fitted formulas that can be easily projected out for any size of Tak board, pretty much indefinitely:

tak stats 2

I told you the math was pretty! A power curve and an exponential curve tell you how many pieces you have to split between the players and the number of capstones each player gets. If you use my numbers from the first projection chart (the one where I didn’t round the number of pieces) you can easily calculate the base formulas yourself using Excel, Google Sheets, or something similar. I’ll happily answer questions if you have them.

So there you have it! Now you can play Tak on any size board you could possibly want to! Of course, whether it’s a good idea to play a 20×20 game of Tak (or  bigger!) is a completely different question. With 225 pieces and 8 capstones per player, that’s going to be one long game. That said, it’d definitely be an interesting one!

One last thing: While I’m sure most people will understand this implicitly, I’m going to state it for the record: Tak is owned by Cheapass Games. It was created by James Ernest and Patrick Rothfuss. I don’t have any rights to the game, nor am I claiming any rights to it. This post is entirely speculative and should not in any way, shape, or form be thought of as official Tak rules. This post is not endorsed or authorized by Cheapass Games in any way, and no offense is meant to them. I’m just a fan of their work who decided to go a little crazy. So there you go. Now go back the Tak Kickstarter! Play the game! Make your own set! Drink while playing the game! It makes it more fun! (Please drink responsibly. This means don’t try making a wooden set using power tools if inebriated, among other things. Don’t be an idiot, m’kay?)

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About Josh

I'm an engineer, brewer, vintner, gamer, hiker, biker, and many other "-ers" besides. I grew up in Connecticut, but now live and work in Delaware. This is where I put various ramblings and musings, as well as tasty recipes for beer. Yes, beer. I brew my own beer, wine, cider, mead, and other fun fermented beverages. It's fun, easy, inexpensive, and I love it. It's something I'd love to do professionally, but that's a plan for a future day. For now I work as a chemical engineer with a bunch of great people that I'm never going to talk about here. Have a great day, and good brewing to all!
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