There is something different about these dice - can you spot it? I'm guessing you'll get it right away ...
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:
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, and1-3-4-6-6-8-9-11. 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 Gamestation - Gamestation.net is likely the original source for the image I used above.
** "Dammit Jim, I'm a statistician, not a combinatrician!"
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| Image found at Chuck-A-Con * |
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)
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
[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!"
5 comments:
I think there is an error in your second die, it should be 1-3-5-5-7-7-9-11. I checked with a computer! The article by Broline has a table of the various different dice, i.e. 4, 8, 12, 20 sides ones.
TPC: You are right! I should have checked it with a computer myself. Thanks for taking the time to set me straight. :-)
Damnit, Jim, DC's a blogger, not a computer checker-outer!
:-)
But I _should_ have checked it, because I am a computer checker-outer too. Once TPC pointed out the bug it only took few a minutes to set up a spreadsheet and check it myself. In my defense, I wasn't feeling well when I worked that out (better now that the antibiotics have kicked in).
I also fixed the Wikipedia link to Sicherman Dice. The mathematical justification there loses me at "cyclotomic polymonials", but from what follows it is obvious that combinatricians [sp?] have some clever tricks up theirs sleeves. It's good to learn new things, and who knows, maybe I will need to factor a high-order polynomial someday. ;-)
I'm sorry, but cyclotomic sounds like something the late Victor Appleton II would have written. "Tom Swift Jr. and His Cyclotomic Randomizer!"
No? Eh, you hadda be there.
And yes, I know VA II is a pen name for a host of ghost writers.
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