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 it 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 16^th-order polynomial**.

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

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:

FOCUS AND SCOPE
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!

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.

BlogRolling

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).

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]

Proof

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!

Pink Panthers, and a few Dropships

I have a couple of Galactic Knights Carnivorian Panther Class Cruisers that I'm try to finish up and get off my workbench. I've been meaning to post some pictures for a while, and a recent post at Super Galactic Dreadnought prompted me to action.
While I had my camera out, I also snapped some pictures of some AeroTech Union and Overlord class dropships that I have been wanting to show off. With a little help from Picassa, you can see these pictures in the embedded slideshow below, or you can see the full size images in my miniatures gallery.





For the Panthers I employed a non-standard method for painting these. It went something like this:

1. Start with a black primer coat.
2. Drybrush a dark grey, then gradually work up to lighter shades.
3. Accidentally splatter with Pink Paint. Really dump it on.
4. Wash off most of the pink paint, and spread the rest around a bit.
5. Accent with magenta ink.
6. Highlight with Metallic Inks, as shown here.
7. Apply matte varnish.
8. Apply a flat black glaze to "deep" areas to further knock down shininess and bring up the contrast.
9. Add lettering in white ink for the ship names (in process).
10. (next) Mount on Ninja Magic flight adapters?

Since these are "Panther" cruisers, I decided to name them after cats I have known. The first is "Telstar" after a big black cat we had when I was very young. The second is "Boris" after an even bigger black* Maine Coon who has assisted in the development of Squadron Strike (Boris allows Ken to type, sometimes). 

* Boris informs me he is actually a mackeral tabby.

The AeroTech dropships were an experiment in how much I could do with metallic inks, and I am very pleased with the result. I started with a black primer, and applied a number of thin layers of silver ink until I got a solid tone. Some highlighting with copper ink, black ink on the doors and some black-lining the details finished them off nicely. These came out with a much better metallic look than I get from metallic paints.

Jane McGonigal on The Colbert Report

Jane McGonigal on the Colbert Report

McGonigal is plugging her new book, Reality is Broken. I just added that book to my wish list too.

More McGonigal!

UPDATE: Gamasutra has a short excerpt from McGonigal's new book.