20 August 2011

The Grinder

Assorted bits of internet, selected from a stratified sample, sorted in ascending sequence, collated, assembled in logical order, then just sort of thrown in a pot and given a good hard shake. Completely by accident, I seem to have a fine selection of game design articles for this edition. How did that happen?

--- The Grinder - August 2011 ---

PAXsims - The latest Game Design blog I am following. Some good stuff here I need to check out.

IO9: Robot Art Wallpapers

A Fafnir crashes the party

Wargamer's Notebook: The Moment  -- "I wish this was something I experienced more often. The point during a wargame after several plays or turns when things magically click. The moment when the scaffolding of the the rules and latticework of the bits fall away and the narrative zooms into the foreground and you - as the player - are completely absorbed by the game. Events occur that could not have been imagined, but that are totally plausible. Victory hangs in the balance. And you are not moving counters or playing cards, but making choices that give you hope of victory."

Space Disaster!
Genius Sir!
More irreverent humor to be found at Perry Bible Fellowship.

Desert Scibes brings us a bit bit of gaming history at Supergalactic Dreadnaught:
NOTE: Scott R. Spicer (credited as "S.R. Spicer, Lt., TFSF"), along with his father, Ron Spicer (R.E. Spicer, Lt. Commander, TFSF), developed the Starfleet Warsminiatures line for Superior Models. As you probably know if you've read this blog before, the SfW universe contains five starfaring factions, each with a distinctive design style. Curious about the origin of this game and its minis, I emailed Scott Spicer about his work on these spaceship models. He was gracious enough to reply to my inquiry, and his response follows in its entirety ...

Make your own custom dice (with a bit of work).

From SMBC: (If only it were this easy!) 

Gamasutra: Game Design Essentials: 20 Real-World Games

Chess, Go, and Life. Which one is the Best. Game. Ever. A nice bit of gamey mathiness from Patterns in the Void.

Finally, this is just cool.

25 July 2011

Taking a RISK - the distribution of armies lost

I came across the following question at boardgames.stackexchange.com:
How can I estimate my chances to win a Risk battle?
Elliot Avedon Virtual Museum of Games, Courtesy Canadian Museum of Civilization.
Now there is already plenty of material on the web about probability in the popular boardgame RISK*, but maybe I can add just a little bit more.

Most people already know that, given the choice, the Attacker should always 3 dice and the Defender should always roll 2, since this always gives the best results (a Nash equilibrium). Since it's always the Attackers choice to roll an attack or not, the relevant question seem to be "How many armies [X] will the Attacker lose if they make [N] attack rolls?"

Using some probabilities from a RISK FAQ for the probabilities of losses, I calculated the average attacker losses (about 0.921 per roll) and standard deviation (~0.81). It's not possible to lose 0.9 armies in RISK! as the attacker losses vary between 0, 1, and 2 per roll. However, as the results of many attack rolls are added up, the losses will begin to resemble the normal distribution. We can plug the average and standard deviation into a normal approximation formula, and get back a probability for losses in a fairly simple calculation. For a given number of attack rolls "A" and number of attacking armies lost "K",
Z = (K - 0.921*A) / (0.811*sqrt(K))
where Z is a standard normal variable (mean 0, standard deviation 1), and the probability of K-or-fewer losses in A attack rolls can be evaluated with the standard normal probability CDF function, otherwise known as the NORMSDIST(Z) function in Excel. That's pretty much it, except maybe for some graphs to show off the results.

Cumulative probability of K attacking armies lost in 25 attack rolls

The blue line shows the cumulative probability of K losses in A attack rolls (vertical red line). The yellow triangles show an approximate 50% confidence interval, meaning that your actual losses should be within this range 50% of the time. The red diamonds show a 90% interval for the same. These intervals are actually slightly wider than the stated 50%/90%; because I rounded outwards to the nearest whole number of armies, and there is no other good way to do it.

Cumulative probability of K attacking armies lost in 35 attack rolls

Recall this is an approximation, and it depends on there being lots of independent random events (dice rolls!) for the approximation to work well. It should start to work very well somewhere between 20-30 attack rolls, and depending on how fussy you are may give usefully accurate results for as few as 10-15 rolls. For smaller battles, or deciding whether or not you should attack just one more time, you might consider more accurate methods (look here, for starters).

Cumulative probability of K attacking armies lost in 70 attack rolls

You can think of this in terms of Defender losses too. Two armies are lost with every attack (between the attacker and defender), so if there are A attack rolls and K attacking armies are lost, that means the defender will be losing 2*A - K armies. For instance, in 25 attack rolls a total of 50 armies are lost; the probability of the attacker losing 20 armies the same as the probability of the defender losing 30.

Cumulative probability of K attacking armies lost in 50 attack rolls

Nostalgic Tangent: Calculating the probabilities of losses for the attacker and defender was one of my first mathematical efforts to figure out a game. I didn't know how to do the calculation, but I wrote a program on my Apple II+ to roll lots of electronic dice for me, and calculated the probabilities that way. Later I learned that this technique is called Monte Carlo simulation, and statisticians do this regularly to examine the properties of new statistical methods.

*** There WILL BE a link to the spreadsheet for these calculations, but it's getting late, so I'll have to add that in tomorrow. ***

Having written this, I realized that I haven't quite answered the question. I've given the probability for a given number of attacks/losses, but the question is "How many armies will it cost me to win?"

There is another approximation to answer that, but it's much less well known. I guess I'll have to write a part 2. Stay tuned!

* RISK is a registered trademark of HASBRO, Inc., of course.

Footnote: The RISK! game images used in this post are from Elliott Avedon's Virtual Game Museum, and are used with permission of the Canadian Museum of Civilization. The Virtual Game Museum has much more interesting game related information, and I may be posting about it again.

04 July 2011

The Grinder - July 4th Edition

A collection of red-glaring rockets and bombs bursting in air, without the rockets and bombs.

The Grinder for 7/4/2011:

 Paint-It-Pink : Ashley has some Battletech math going on --> How I Learned to Stop Worrying and Love the Medium Laser. A good discussion of how to evaluate the relative strength of weapons in Battletech, or any game.

"Can I haz tactix?" 
Found on Operation Odyssey Dawn: If the animation doesn't work, go see it here.

Linkback! --> 程阳:Probability versus Odds

MathOverflow: Which popular games are the most mathematical?

Proof: The 120 cell is a 4 dimensional figure that can be considered the 4 dimensional analog of the dodecahedron. It has 720 five sided faces, 1200 edges, and 600 vertices. This animation shows 3 dimensional cross sections of the 120 cell in a way that is similar to taking 2 dimensional cross sections of a 3 dimensional figure. Translation --> Very Cool animation!

The Number Warior: Q*Bert Teaches the Binomial Theorem (an award winner too). Sort of a long (2-part) video.

Proof-of-False: Do games offer a solution for US Tax Reform?

From doctormattA Collection of Dice Problems with solutions and useful appendices
Mike Reilly is a Toy/Puzzle designer and screenwriter, see what he has done at Reilly4Puzzles.

Reinwood's CBT Workbench gives us an AAR for Fourth Succession War: Skondia The Final Battle Kublacon. AND it's got no math in it. Honest!

A small update to my Graph Paper Race post (added link to a relevant article). This continues to be one of my more popular posts.

I didn't plan this, but somehow this has ended up being the most math-heavy edition of The Grinder to date. Oh well, it's all sausage now.

03 July 2011

Dice Distributions Revisited

Recent thoughts about calculating the distribution of the maximum sum of several dice (ex: roll 3, sum the highest 2) made me realize I needed a better tool for calculating the distribution of sums of dice in the first place. I first wrote about this some time ago in Dice Distributions, so I knew how to do it better, I just hadn't gotten around to doing it.

Tangent: While researching this I can across a great set of mathematical Dice Problems from Doctormatt (Web Page, Blog).

And now back to our story -

I first set up a spreadsheet to give me Pascal's Triangle, which looks like this:

This gets used in lookup functions to calculate (in a second worksheet) what I'm calling the "Dice Triangle", The number of dice [N] rolled is indicated in the column headers, and the sum of N D-sided dice [X] rolled in indicated in the first column. The number of ways to roll that sum is indicated in the table. The number of sides on the die [D] can be changed by entering a different value into the green shaded cell.

The first D rows of the table come directly from Pascal's Triangle. Subsequent rows are calculated from previous rows of this table. A few more details of how this is done in my earlier post (Dice Distributions), otherwise you can ask me, or dig into the spreadsheet for yourself (sorry, in a hurry today).

Here is the spreadsheet: Dice Distribution Calculator
I have not made this public, so you cannot change it directly online. You can download a copy for yourself (under the File dropdown) and play with it to your hearts content.

01 July 2011

Maximums and Minimums of Dice Rolls

A week month a while back I received questions from Christian and a read post from Saxywolf on essentially the same question: What is the probability of rolling a given value on an D-sided die, if you roll N dice and take the highest (or the lowest).

Here's the trick:
The probability of rolling a 1 on 1 d6 is 1/6. (regular 6-sided dice)

For 2d6 the probability of rolling a 1 as the maximum is 1/6 times 1/6, or 1/(6*6) = 1/36.
For Nd6 the probability of rolling a 1 as the maximum is 1/6 times itself N times, or (1/6)^N.

For D-sided dice just substitute D for 6 above, so that the probability of rolling a 1 as the maximum of N D-sided dice is 1/D times itself N times, or (1/D)^N.

Now consider the problem of rolling 2-or less as the maximum. The probability of rolling 2-or-less is 2/D, and the probability of rolling a  2-or-less as the maximum is 2/D times itself N times, or (2/D)^N.
That's 2-or-less, but we really just want the probability of rolling 2, not 1 or 2.  BUT we already know the probability of rolling a 1 as the maximum on the same dice, so we can subtract that to get what we want:

The probability of rolling a 2 as the maximum is 2/D times itself N times, minus the probability of rolling 1 as the maximum, or (2/D)^N - (1/D)^N.

And that's it. Using the same math you can work the complete distribution of the maximum for any number of dice with any number of (equally likely) faces. Just start at 1 and work up. For minimums, just turn the problem around and find the probability of X-or-less.

Still too much math? Fear not for there is a spreadsheet to do the calculations for you:
 Link to Google Docs Spreadsheet

See the blue numbers in the green-shaded cells? Change those to the number of side on your dice and the number you want to roll, and it will calculate the distribution for you. It might even work inside the blog?Nope, but it was worth a try. The spreadsheet is now public and can be edited at the link above (also downloaded). Changes made there WILL show up here when the page is reloaded, which means you are looking at whatever was most recently entered.

And a chart to display the results:

This is a bit of an experiment, both linking to a shared spreadsheet, and adding the HTML code to it directly inside my blog post. One upshot of this is that when one person changes the spreadsheet, it will change it for everyone. Play nice! Let me know if it works too. [Fixed!]

04 June 2011

The Grinder

[A dazzling display of delightful de ... um ... I need a D-word ... deviations ... detritus ... de-links?

The Grinder -  6/4/2011 edition

And speaking of dazzle, could Dazzle-camouflage make a comeback? This recent research supports says Dazzle Camouflage Affects Speed Perception. [Hat-Tip IO9]

Image: WWIaviation.blogspot.com
This just in! Check out some dazzling WWI aviation paint schemes.

Terrain table pictures ... Shiny!

This could be interesting ...
Invasion3042 is a massive multiplayer online game that is based off the game Battletech. It is a free game and is not for profit.
Has anybody tried it?

Discoblog brings us Tiny Toss-able Robots.

World Peace Games, with teacher John Hunter. Video from TED. It's a bit slow to get started, but gets interesting about 8 minutes in. Never cross a 9-year-old girl with tanks!

[Hat-Tip Greg Laden]

[The Endeavour] There are exactly five platonic solids*, and you can prove it!
* Perhaps more familiar to my readers as dice - the d4, d6, d8, d12, and d20.

Another video, this one with singing and dancing! Roll A D6.

[Hat-Tip: Operation Odyssey Dawn]

[IO9] MTG as an RPG?
The was an MTG computer game, long ago, with very nearly this premise. With a bit of creativity it could be good for multiplayer too. Before that was a great game called Master of Magic. Also, there could be a new Star-Trek animated series!

Play JAM! Can you beat the computer? Can you beat it every time?? Can you figure out the secret??? (without peaking!) Here is a hint - You have almost certainly played this game before, and many times. [Hat-Tip Terrace Tao]

Non-Transitive Dice

Starcraft Humor from Abstruse Goose. I didn't get it until I saw the caption.

Green Cube: The Physics Boardgame

Enough bedazzlement for two sittings, but that's happens when I don;t post for a whole month. Writers-block sucks. Want to preview the next Grinder, before I post it? Check out my Google Reader Shared Links page for GRB.

27 April 2011

Doug Chaffe

Update [7/14/11]: Catalyst Game Labs is hosting a sale of Doug Chaffe original artwork at Gencon 2001.

Randall Bills writes about the passing of Doug Chaffee. Better go read that first. Here is Randall's opening paragraph.
In 1994 I walked into a game store and beheld BattleTech’s CityTech Second Edition on a shelf. The cover was by an artist I’d not seen before, but it was action-packed, evocative, and wonderfully done…everything a fan could ask from a cover. Of course I left with it under my arm. For the next 15 plus years Doug Chaffee would be an indelible part of crafting the visuals for BattleTech.
I confess I didn't know anything about Doug Chaffee before this, but Battletech players should know his work on sight ...

Image: Chaffee Studios
... because it is nothing short of iconic. Much more to be found at Chaffee Studios.

Here are a few links that my be of interest:
Battletech Wiki: Doug Chaffee
Book Illustrations

The Grinder

\                                                                                                                                                             This edition of The Grinder is brought to you with the assistance of Darwin the cockatiel, who inserted the backslash at the beginning of this post, and bit at my fingers as I type this. I haven't been posting much lately, but I have been writing, and hopefully some of that will find its way back here soon.
Now on with the Grind.

Some fun with Game Theory on the British show QI (short for "Quite Interesting").

[Via Terrence Tao's Buzz]

Airships for the 21st Century
Which has nothing to do with games or math; I just think it's cool.
[Hat Tip David Brin on Twitter]

Mo Rocca visits Gen-Con and plays boardgames: Video at CBS News.

Is it just me, or does everybody have friends that ask them questions about counting problems and combinatorics? (I'm looking at YOU, RR!)
Somehow, I think it's just me.

Ian Schreiber writes about education and games at Teaching Game Design: My Problem With Gamification.

 The Last Cause is a movie in pre-pre-production, but it has mechs and clones, so Battletech players are likely to take notice.
Edit (2016): This early Kickstater success is now considered on of it's biggest early failures. Oh well.

Game Designer, Graphic Designer, Wargamer Extrodinaire, and a bit of gaming history I should probably already know, but somehow didn't: Redmond Simonsen. Go read about him. [Hat Tip 2 Grog News]

04 April 2011

Fletcher Pratt's Naval Wargame

John Curry, editor of the History of Wargaming Project, has a new book out:

I first wrote about Fletcher Pratt's Naval Wargame almost two years ago in The Origin of Battletech. At the time a new edition of the book was still in the works, so I waited to get it. This edition includes some previously unpublished material from some of Pratt's original players and umpires.

My copy is on order, and I'm looking forward to seeing it soon. Maybe I will organize a local play session? 

02 April 2011

Sicherman's Dice

There is something different about these dice - can you spot it? I'm guessing you'll get it right away ...
Image found at Chuck-A-Con *
You can't see the non-facing sides, but the d6 on the left is labeled with 1-2-2-3-3-4, and on the right labeled with 1-3-4-5-6-8 (like this). That's not our standard 1-2-3-4-5-6, and if someone rolled these on the gaming table the 8-pip is a dead giveaway that something is off.

Now here's the trick: The probability distribution for the sum of these Sicherman Dice is identical to the distribution of the standard 2d6, so if you only see the results (the sum) there is no difference at all.

The mathematics for this gets into Generating Functions and Combinatorics, but essentially is involves doing the algebra to show that:

(x + x2 + x3 + x4 + x5 + x6)2 = (x + 2x2 + 2x3 + x4)(x + x3 + x4 + x5 + x6 + x8)

Where the left-hand-side is the generating function for the sum of two standard 6-sided dice, and the right-hand-side is the appropriately factored generating function for the sum of Sicherman's dice. (OK, maybe a little harder than that.) There is only one way of doing this with 6-sided dice, but such variations exist for other polyhedral dice. It seems to be possible in general to do this with N-sided dice, and there might be multiple ways of doing this for some. The Mathematics Magazine article "Renumbering of the Faces of Dice" by Duane Broline (1979) goes into some detail, but I cannot access the full article from home. If I can grab it at work maybe there will be an addendum.

The Hard Way

I tried this working out possible numberings for Sicherman-type 2d8 dice by scribbling with pencil and paper until I found a combination that worked. On my third-and-a-half attempt I came up with 1-2-2-3-3-4-4-5, and 1-3-4-6-6-8-9-11. CORRECTION: TPC checked more carefully than I did, and offers 1-3-5-5-7-7-9-11 in place.  It took me a while to work this out "the hard way", but it was probably still faster than I could have factored a 16th-order polynomial**.

[Hat-Tip to The Endeavor/John Cook. Again!]
[As seen on Eon.]
Sicherman Dice are available from Amazon, or directly from GamestationGamestation.net is likely the original source for the image I used above.
** "Dammit Jim, I'm a statistician, not a combinatrician!"

26 March 2011

Fun with Physics Special Topics

I recently discovered the online journal Physics Special Topics (current issue, archives). These are essentially collections of student articles on various topics. Here is how the journal describes itself:

This is an undergraduate journal for year 4 (MPhys) students in the Department of Physics and Astronomy at the University of Leicester. The journal accepts brief papers on topics original to the authors. It does not accept reviews or summaries of other peoples work. It is managed by an editorial board which rotates round the student body overseen by a member of staff. The journal forms part of the assessed element of the MPhys degree. Assessment is by number and quality of accepted publications and referee reports.

The students are writing serious article on a wide variety of topics, some of which are inspired by by games and science fiction. As science goes, these is not exactly cutting edge, but it sure is fun. Some of the articles that caught my eye are:

How Heroic is a Hero? An exploration of the probability distribution behind character creation in the D20 system.

Niven Rings looks at some of the physics of Larry Niven's Ringworld.

Can superhuman muscles stop bullets?  Ask the question, "Just how bulletproof is Superman, anyway?"

And much more.

This is perhaps slightly off the usual theme of this blog, but surely playing games and reading SciFi has raised a few of these questions in the minds of gamers - "Just how what that work, if we actually tried it?" - Like many things, the fun is in the finding out. 

If one of these articles catches your interest, please post about it below. Have fun!

07 March 2011

A few more minutes in the library

I received the following email, apparently from a young person, so I won't reveal the name:
Hi im 14 and ive loved modeling since i was 10. I have 3 dioramas Railroads Cut, Gettysburg, Pennsylvania/ The Battle of the Hurtgen Forset, France/ and one that i made up and created when i was 11 Bloody Ridge, Kentucky. I am know working on the battle of waterloo and was wondering do you have any tips for me i have already painted 80 bristish and prussian and 15 French calvalry and am working on the french foot soldiers. I am hoping to get a large board than i have know but any terrain tips or details?

There are many sites on the internet where you can find such advice, but I have a different suggestion: Go to your local library and look for books on "military modeling", "model railroading", "Civil War history", and anything you can find by Sheperd Paine. Browsing those shelves of related hobbies can show you more in a few minutes than you will find in hours of Googling the internet, because library shelves aren't sorted by keywords. 

That doesn't mean you won't find the same information with your favorite search engine, but it's hard to search for ideas you don't know exist. Look in your library - because libraries are absolutely full of good ideas. You will be pleased with the results.

This is good advice for anyone who enjoys games, miniatures and modeling. DO NOT limit yourself to learning only from others within a narrow segment of your hobby. Reading some of the bulletin boards, you might get the impression that painting miniatures is nearly exclusive to just one or two games, and nothing could be further from the truth. Make a serious effort to study what others are doing, and bring back what you learn to make your own hobby work even better.

14 February 2011


I've been making some long overdue repairs to my Blogroll, adding folks who link to me, and weeding out some feed of less interest or relevance. If you look on the sidebar you will actually see 4 blogrolls - I've built up so many that I needed to organize them a bit. There are loosely organized groups for Battletech, Related math, Friends and interesting stuff, and Game Design. If you want a link let me know, especially if you link to me already and I missed it (I'm sure there are a few).
GBR Giant Battling Robots Favicon

08 February 2011

The Grinder - Valentine's Day Edition

Links for the one you love, ground up and arranged in a nice bouquet.

The Dot and the Line: A Romance in Lower Mathematics
From Norton Juster, and the Immortal Chuck Jones

[IO9] - Mathematicians figure out how to fend off gold-diggers – with game theory  ---  This should be probably be re-titled: Two male Mathematician's figure out yet another way not to get girls.

Spiked Math
Spiked Math has an experimental simplified viewing page, and a new Spiked Math app for the iPhone/iPad.

[The Fallacy Files] - The Puzzle of the Absent-Minded Professors
Because no one can ever have enough logic puzzles.

That's a Terrible Idea: Actually has some pretty good idea's, and I have added them to my blogroll under Game Design Blogs (somewhere down there, but hey, a link is a link). Here are two good samples:
  1. Games from the Ground Up
  2. The Role of Chance

Dancing Robots! The party really gets going about 3 minutes in.
[Found on Engadget, via Pharyngula]


More goodness from Proof: Snowhedron

[Dark Roasted Blend] If you like the artwork for Catalyst Games Labs new game Leviathans, check out The Great Eastern, a leviathan steamship on the Victorian era.

[Discoblog] Can video games make smarter spies? Oh come on guys, get Sirius!

That's all for this edition. Be sure to set aside the games you love for a little time with the one you love. Or better yet, get them to play a game with you!
GBR Giant Battling Robots Favicon