36 sided die? Replacing two 6 sided dice. Where to buy?
Arthur Gibbs
that would show up as if you rolled two six sided dice? I know this sounds
lazy. But we play settlers and other games that require adding two six
sided dice. Sure would be neat to have one 36 sided die. One that had, one
2, two 3's, three 4's, and so on. So that you still only get numbers 2-12
as you would rolling two 1d6.
Anyone know if this even exists and if so where I can buy one?
Thanks in advance!!!
Arthur :)
Jae
says...
There is a company that will mold "any sided dice" for you.
I forget who it is though.
BL
red six-sider encased in a clear plastic six-sider that has black pips on
it. Roll once, get two results (you'll still have to do the math of adding
them together). Not exactly what you're looking for, but worth a look if
nothing else pans out...
/Brian
"Arthur Gibbs" <nos...@aol.com> wrote in message
news:3GAa7.84260$TM5.10...@typhoon.southeast.rr.com...
Jasper Phillips
I've recently seend a bunch of transparent hollow 6 siders, with another
die inside. These are pretty cool, and similar to what you're talking
about; well, except that it's not a die with 36 sides. ;-)
-Jasper
Tracy Johnson
Just put this into a cell and press f9 every time:
=TRUNC(RAND()*6)+1+TRUNC(RAND()*6)+1
Jae wrote:
>
> In article <3GAa7.84260$TM5.10...@typhoon.southeast.rr.com>, Arthur Gibbs
> says...
> >
> >Anyone know of a place I can buy one 36 sided die? One that has the numbers
> >that would show up as if you rolled two six sided dice? I know this sounds
> >lazy. But we play settlers and other games that require adding two six
> >sided dice. Sure would be neat to have one 36 sided die. One that had, one
> >2, two 3's, three 4's, and so on. So that you still only get numbers 2-12
> >as you would rolling two 1d6.
--
BT
NNNN
Tracy Johnson
Justin Thyme Productions
Sponsors Free Multiuser Wargaming on the WEB at:
http://hp3000.empireclassic.com/
Gilbert Benoît
sided polyhedron. The fact that the ten sided die only has axial
symmetry has always bothered me (call me a purist!).
Gilbert Benoît
Ottawa, Canada
Rich Shipley
news:3B6AF8FA...@home.com...
> Also, keep in mind that there is no such thing as a regular thirty-six
> sided polyhedron. The fact that the ten sided die only has axial
> symmetry has always bothered me (call me a purist!).
There are dice that have more than axial symmetry, but don't have regular
faces. I have some 30-siders (rhombus shaped faces) and I've heard of
34-siders.
Rich
Smeagol
You may be a purist, but you are also absolutely right! A ten sided die
has only axial symmetry, and that is the only way it can be done to
insure that the chance for turning up on any side is the same for every
one of those sides.
Is there no way for a 36-sided die? I'm not an math-wizard, but wouldn't
it be possible to have a top layer of 6 triangles, connected to a "band"
of 12 same sized triangles, and this copied for the other half of the
die?
By the way, wouldn't it be great to have a 7-sided die? ;-)
Jeroen.
David desJardins
> Also, keep in mind that there is no such thing as a regular thirty-six
> sided polyhedron. The fact that the ten sided die only has axial
> symmetry has always bothered me (call me a purist!).
What you say is not true. The symmetry group of the ten-sided die
includes both axial symmetry and reflection through the central plane.
A die without the latter symmetry wouldn't be fair. (Imagine two
five-sided pyramids on the same base, but one of the pyramids is much
taller than the other. This die has axial symmetry, only.)
I think a "purist" is happy with any die for which the symmetry group is
transitive on the faces: this is sufficient to imply a "fair die".
Asking for the die to be regular (i.e., the symmetry group is transitive
on the vertices also, and the faces are regular polygons) seems an
arbitrary requirement of only aesthetic value.
David desJardins
Justin Green
> that would show up as if you rolled two six sided dice? I know this sounds
> lazy.
I can't help you, but here's a machine that will roll two dice every
2.5 secs (34560 rolls per day...or approx. 450 games of Settlers). It
takes a picture of the dice after each roll and adds the numbers
together for you, so you don't have to worry about THAT. You'd
probably have to hire a Canadian physics grad student or two to work
and maintain it...but they don't cost much. ;-)
Christopher Bourassa
difficult? Or is the mathematics of adding numbers too extreme once you reach
double digits? Did you stop to think that a thirty-six sided die might be
heavier than two six sided dice? In that case, you would start to complain.
Maybe you would have to find an alternative plan.
I'm sorry if this come off as being incredibly sarcastic, but rolling two dice
is as simple as possible.
Gilbert Benoît
David desJardins wrote:
>
> I think a "purist" is happy with any die for which the symmetry group is
> transitive on the faces: this is sufficient to imply a "fair die".
> Asking for the die to be regular (i.e., the symmetry group is transitive
> on the vertices also, and the faces are regular polygons) seems an
> arbitrary requirement of only aesthetic value.
>
What are games about, if not aesthetics?
Gilbert
Dan Becker
>
> Also, keep in mind that there is no such thing as a regular thirty-six
> sided polyhedron. The fact that the ten sided die only has axial
> symmetry has always bothered me (call me a purist!).
>
And don't forget you can make fair dice of any arbitrary number of sides
by using a cylinder with flattened faces. For instance, pencils with
marked sides make great dice. So do coins. In fact, the more faces, the
easier these things roll and the better they randomize.
This would be axially symmetrical only and might not satisfy aesthetes.
Speaking of aesthetics, my American Heritage dictionary uses "cubic" or
"cube" in all definitions of the word dice (except "1. Pl. of die"). By
this definition, 12-siders, 20-siders, non -regulars, and non-6-siders
would not be dice.
My dictionary says the origin of die is "OFr de". Does anyone know more
on the origin of this word?
--
Thanks, Dan
email mailto://beck...@io.com
web http://www.io.com/~beckerdo
John R. Cooper
wrote:
(Shameless Plug Warning)
If you have a Palm device, I invite you to check out DicePro. It
will do what you want (along with a hundred other dice rolling duties
you'll probably never need), and best of all, it's freeware!
http://www.geocities.com/TimesSquare/Realm/9565/
Cheers,
- John
Gilbert Benoît
Dan Becker wrote:
>
> Speaking of aesthetics, my American Heritage dictionary uses "cubic" or
> "cube" in all definitions of the word dice (except "1. Pl. of die"). By
> this definition, 12-siders, 20-siders, non -regulars, and non-6-siders
> would not be dice.
>
So dictionary writers do not play D&D? That is hardly a surprise! ;-)
> My dictionary says the origin of die is "OFr de". Does anyone know more
> on the origin of this word?
>
Well, "dé" is the French word for die (i.e. not just in Old French).
I do not know the etymology, though (since the Latin word was "alea",
it came from somewhere else).
Gilbert
Matthew Baldwin
Bourassa wrote:
> I don't think your sarcasm helps. You could have just ignored his
> [request] if you didn't really want to help.
Seconded.
Mike Schneider
In article <3B6AF8FA...@home.com>, Gilbert =?iso-8859-1?Q?Beno=EEt?=
<dwork...@home.com> wrote:
> Also, keep in mind that there is no such thing as a regular thirty-six
> sided polyhedron. The fact that the ten sided die only has axial
> symmetry has always bothered me (call me a purist!).
I don't mind that so long as they're the type that look like cubes when
viewed from a certain angle (they group nicer on the table and have a
better tactile feel). But at least you can be certain the probability for
all faces is the same (baring imperfections), unlike with those wierd
7-sided dice I've seen, which are a fat pentagonal column (hey math
whizzes: how *do* you prove that (or make) a polyhedron with different
types of facets will have the same probability of landing on each? I know
there's more to it than simple surface area).
--
http://groups.yahoo.com/group/American_Liberty/files/al.htm
Reply to mike1@@@usfamily.net sans two @@, or your reply won't reach me.
David desJardins
> (hey math whizzes: how *do* you prove that (or make) a polyhedron with
> different types of facets will have the same probability of landing on
> each?
I think you can't prove that. The relative probability of the different
faces will depend on how you throw it.
David desJardins
Christopher Bourassa
Did you go through every post looking for that? :)
Jason E. Schaff
>
> Is there no way for a 36-sided die? I'm not an math-wizard, but wouldn't
> it be possible to have a top layer of 6 triangles, connected to a "band"
> of 12 same sized triangles, and this copied for the other half of the
> die?
The problem with thus proposed geometry is that the die would have a
disproportionately high probability of ending up on one of the "band" faces
as oposed to one of the top faces. While sitting on a top face the die
would be on a less stable footing and would tend to tip over onto a "band"
face.
I would think that there must be some geometry that would produce a "fair"
36-sided die though. I have actually seen a 100-sided die for sale once:
the thing was _huge_, about the size of a plum.
Jason
(stunned that I actually managed to put my physics and group theory classes
from college together and use them for a real-world problem!)
--
---------------------------------------------------------
Jason E. Schaff
Remember always to laugh, for
such is the true sound of humanity.
-- unknown
---------------------------------------------------------
Mike Schneider
<jv....@nospamquicknet.nl> wrote:
> You may be a purist, but you are also absolutely right! A ten sided die
> has only axial symmetry, and that is the only way it can be done to
> insure that the chance for turning up on any side is the same for every
> one of those sides.
> Is there no way for a 36-sided die? I'm not an math-wizard, but wouldn't
> it be possible to have a top layer of 6 triangles, connected to a "band"
> of 12 same sized triangles, and this copied for the other half of the die?
36 factors into 1,3,4,6,9,12 and 36.
The problem is that by altering the angles and number of verticies at
various facet intersects, you'll alter the probability that some triangles
will be landed on versus others, even if they're all identically-shaped (I
don't think that would even be possible in this cast, since you need six
"pointy" triangles meeting at the "top" and "bottom" of the die and also
around its "equator", while the "shoulder" and "knee" zones would have
five-point vertices. Screw the probability -- I'd just like to see an
example of one.
Alternatively, you could take your 12-sider (pentagon facets) and find a
way of trisecting the pentagons such that each of the three new facets
have an equal probability of being landed on.
Or take your normal d6 cube and facet each square into six parts of
equal-probability (it'd work, after much tricky math, as per above, but
certainly be uglier-looking).
Possibly a platonic solid (or truncated derivative) could be further
truncated to create a 36-sided solid. If anyone has an URL to a visual
"encyclopedia" of such solids (or word of any software manipulators), I'd
be very grateful. Otherwise, you're probably stuck making an ugly "barrel
roller" type of dice if you want to ensure probability without any complex
mathematics.
> By the way, wouldn't it be great to have a 7-sided die? ;-)
You mean besides rolling a d8 and just re-rolling any 8s?
bdy...@network.boxmail.com
Check out http://hjem.get2net.dk/Klaudius/Dice.htm for a discussion of
proper dice and a table of shapes for various dice. The largest you
can get without doing the two-cones method is 120, and there aren't many
others.
--
Brian Dysart | The RNG giveth, and the RNG taketh away.
bdy...@network.boxmail.com | "...and eight for the fruit bat."
www.rahul.net/bdysart/ | <*> Code Code block: C---
David desJardins
>>> (hey math whizzes: how *do* you prove that (or make) a polyhedron with
>>> different types of facets will have the same probability of landing on
>>> each?
>
> proper dice and a table of shapes for various dice.
Mike Schneider asked about dice with different types of faces. The page
you refer to lists dice with only one type of face.
David desJardins
Kent Reuber
Chessex makes a D34, which is number 1-34. No, really:
http://www.chessex.com/Special_d34.htm
Deck Dice is kind of what you're looking for. This is a set of 36 cards
that have the numbers 2-12 in the proper ratios for a 2D6 distribution.
http://www.io.com/~sos/bc/deckdice.html
--
Kent Reuber (reu...@stanford.edu) Phone (650)725-8092
Senior Networking Specialist Fax (650) 723-0908
Networking Systems 241 Panama St., Pine Hall
Stanford University Stanford, CA 94305-4122
Llarry
faces (numbered 6 and 7). 1 through 5 appear across the edges connecting
the points of the two pentagons, so as to come out clearly on top when one
of the 5 "sides" lands on the bottom. Rolls pretty fairly - the key was
balancing the size of the sides with the faces.
--Llarry
"Smeagol" <jv....@nospamquicknet.nl> wrote in message
news:shDa7.6267$w91.1...@news.quicknet.nl...
George W. Harris
<jason...@home.com> wrote:
:I would think that there must be some geometry that would produce a "fair"
:36-sided die though. I have actually seen a 100-sided die for sale once:
:the thing was _huge_, about the size of a plum.
The only geometry that produces a "fair"
36-sided die would be one which consists of two
18-sided (where the ones at Giza are considered
4-sided) pyramids base-to-base. One could also
produce a similar die with the one pyramid rotated
10 degrees and the faces transformed from icoseles
triangles to kites, but such a die would land with an
edge, rather than a face, up.
The 100-sided die sold by Zocchi is not fair.
:Lawson Chris <ChrisR...@icl.com>
--
They say there's air in your lungs that's been there for years.
George W. Harris For actual email address, replace each 'u' with an 'i'.
Tracy Johnson
Of course, just look at Appendix L of the RPG game of Hackmaster.
I shows you how to "prime" the dice with a 'fame rub' by passing
them over the authors signature and the 'emergency purge' by
blowing on the dice.
Don't forget there was a thread a few years back about the
cohabitation of 'dice lice' in the dice recesses that may
affect the weight of the dice on one side or the other.
Eric Haas
wrote:
Chessex has the 34-siders. They're basically two 17-sided pyramids,
base-to-base.
Mike Schneider
That's like saying that because it's possible to flip a quarter in a
certain manner as to produce more heads than tails (you pitch it so
wobbles rather than dumbelling end-over-end) that the probability of a
coin cannot be determined. Obviously this isn't the case.
There must be a way of calculating probability with (for one example) a
computer simulation casting a die repeatedly from different angles
(ranging from vertical to extremely oblique) and tallying up the results.
Variables needed for the simulation: mass of the die, velocity, and angle
of impact (by side, edge or vertice); the program would need to be able to
accurately simulate "bouncing" and subsequent kinetic energy dissipation
in a gravity environment. While the math is completely beyond me, I should
think that applying physics to bouncing solids wouldn't be impossible for
someone seriously into the stuff.
Mike Schneider
<jason...@home.com> wrote:
> Smeagol wrote: [in part]
>
> >
> > Is there no way for a 36-sided die? I'm not an math-wizard, but wouldn't
> > it be possible to have a top layer of 6 triangles, connected to a "band"
> > of 12 same sized triangles, and this copied for the other half of the
> > die?
>
> The problem with thus proposed geometry is that the die would have a
> disproportionately high probability of ending up on one of the "band" faces
> as oposed to one of the top faces. While sitting on a top face the die
> would be on a less stable footing and would tend to tip over onto a "band"
> face.
>
> I would think that there must be some geometry that would produce a "fair"
> 36-sided die though. I have actually seen a 100-sided die for sale once:
> the thing was _huge_, about the size of a plum.
D&D 100-siders "cheat": Their just plastic spheres with circular dimples
ground into them. IOW, they're more like golf-balls than platonic solids.
Personally, I wouldn't allow them in any roll-playing campaign I'm
running, because the facets are so small relative to the mass of the die
that suble imperfections in the composition of its construction coupled
with the less-than-perfect arrangement of the dimples amplifies the
probability of distorted results. Two tossed ten-siders should yield far
more even results for all numbers.
Mike Schneider
<tghl...@qwest.net> wrote:
> I have a pair of 7-siders I picked up at a con. Two parallel pentagonal
> faces (numbered 6 and 7). 1 through 5 appear across the edges connecting
> the points of the two pentagons, so as to come out clearly on top when one
> of the 5 "sides" lands on the bottom. Rolls pretty fairly - the key was
> balancing the size of the sides with the faces.
I've seen the 7-sider you're talking about and examined it closely; it's,
as you describe, a pentagonal column cross-section. A pentagon side on the
die has a larger surface area than a side square. I ASSume this was found
necessary to counteract the tendancy (otherwise) of the die to roll along
the column axis (yielding higher probability of landing on a square edge
than a pentagon side if the side and edge facets had the same area). How
they determined what ratio of edge-to-side to arrive at is something I'd
like to know.
Mike Schneider
gha...@mundsprung.com wrote:
> On Fri, 03 Aug 2001 23:17:47 GMT, "Jason E. Schaff"
> <jason...@home.com> wrote:
>
> :I would think that there must be some geometry that would produce a "fair"
> :36-sided die though. I have actually seen a 100-sided die for sale once:
> :the thing was _huge_, about the size of a plum.
>
> The only geometry that produces a "fair"
> 36-sided die would be one which consists of two
> 18-sided (where the ones at Giza are considered
> 4-sided) pyramids base-to-base. One could also
> produce a similar die with the one pyramid rotated
> 10 degrees and the faces transformed from icoseles
> triangles to kites, but such a die would land with an
> edge, rather than a face, up.
As example (a 34-sider): http://www.chessex.com/opaque_d34.htm
> The 100-sided die sold by Zocchi is not fair.
>
> :Lawson Chris <ChrisR...@icl.com>
--
David desJardins
>> The relative probability of the different faces will depend on how
>> you throw it.
>
> certain manner as to produce more heads than tails (you pitch it so
> wobbles rather than dumbelling end-over-end) that the probability of a
> coin cannot be determined.
No, it's not the same. A coin (or die) is fair because, if you launch
it in an unknown orientation (uniform random in SL(3,R)), then the
probability of coming to land on each face is uniform, regardless of the
probability distribution on its linear and angular momentum, and the
orientation and characteristics of the surface it will land on.
Of course, if you launch a fair coin or die in a known or controlled
orientation, the result won't necessarily be fair.
Dice without transitive face symmetry don't have this property. The
probability distribution that they generate will depend not only on
initial orientation, but also on the initial momentum, and on the
surface that the die is rolled on.
> There must be a way of calculating probability with (for one example) a
> computer simulation casting a die repeatedly from different angles
> (ranging from vertical to extremely oblique) and tallying up the results.
Perhaps. But if you get different results at different angles, then
what good is that? Any die that will be "fair" when rolled at one
particular angle, with one particular velocity, on one particular type
of surface, won't be fair when rolled at other angles or velocities or
on other types of surface.
David desJardins
Christopher Dearlove
<de...@math.berkeley.edu> writes
>Gilbert Benoît <dwork...@home.com> writes:
>> Also, keep in mind that there is no such thing as a regular thirty-six
>> sided polyhedron. The fact that the ten sided die only has axial
>> symmetry has always bothered me (call me a purist!).
>
>What you say is not true. The symmetry group of the ten-sided die
>includes both axial symmetry and reflection through the central plane.
>A die without the latter symmetry wouldn't be fair.
Just to nitpick does it work to take a ten sided dice, twist one
end by 36 degrees (half a face) and rework the join with a zig-zag
interface (so each face is a kite)? (I can sort of see this in
my head, but haven't tried to verify it.) If this works then this
wouldn't have exactly reflection symmetry in the plane, but something
else (and as you not a die but must have some such symmetry to be
absolutely fair).
--
Christopher Dearlove
Mike Schneider
<da...@desjardins.org> wrote:
The coin analogy still holds with this argument: If I "roll" one by
pitching it at an oblique angle onto a felt-covered pool table, I can make
it come up heads every time. Ditto craps dice at Vegas (which is why
gambling table manufacturers now put in a "slide bump"). With a
well-designed more spherical die, however, this kind of cheating would be
nearly impossible (since it is more precariously balanced over its center
of gravity) -- even if one type of face might be more likely to appear
over the other from a certain angle of toss.
The purpose of the computer simulation is to determine the optimal ratio
between the surface areas of different facets in order to generate a
smooth probability distribution from all angles of impact, but with strong
weighting toward a 5- to 10-degree angle of impact (or whatever is
determined to be the mean angle of "final bounce").
I would LIKE to think that the creaters of the pentagonal 7-sider did
something like this (but I doubt it).
IAE, Q. for the geometry guys: Is there a truncated platonic solid (or
derivation) with 36 sides?
David desJardins
> The coin analogy still holds with this argument: If I "roll" one by
> pitching it at an oblique angle onto a felt-covered pool table, I can make
> it come up heads every time.
You can only influence the result of a coin flip if you start with the
coin in a known orientation. If you start it in a completely unknown
orientation, then it's equally likely to land "heads" or "tails", just
because it's equally likely to start "heads" or "tails", and the two are
completely symmetric. This is why dice and coins are considered fair:
if you make no *attempt* to manipulate them, or to observe or control
the starting state, then they will generate uniform random results.
That isn't (can't be) true of dice which don't have the same symmetries.
> The purpose of the computer simulation is to determine the optimal ratio
> between the surface areas of different facets in order to generate a
> smooth probability distribution from all angles of impact, but with strong
> weighting toward a 5- to 10-degree angle of impact (or whatever is
> determined to be the mean angle of "final bounce").
>
> I would LIKE to think that the creaters of the pentagonal 7-sider did
> something like this (but I doubt it).
In this case, there's only one degree of freedom (the ratio of the
length to the width). You could make a bunch of samples with different
ratios, and try rolling them, until you come up with one that you are
happy with. This is going to be more accurate than a simulation.
David desJardins
ccs-ed
regarding the pains that New Jersey & Neveda casinos go
thru in testing EACH die they place in service. They have a
little device, very precise, which grabs opposite vertices and
then they spin it. If the weight is not distributed perfectly, it
worbbles. This test would work with cubes and dodecra-
hedrons (which also have parallel opposite faces) but not
tetrahedrons.
Of course, just because the dice are fair, doesn't preclude
a gambler from trying to influence the outcome - hence in
craps, you must keep the dice in view at all times and the
roll MUST first hit the table and then bounce off the wall.
When throwing a die, say a "6" comes up. This means the
die actually landed on the "1" face. The other 4 faces are in
the vertical and there is no confusion as to which face is "up."
Now throw a Titlist golf ball - you know the one with 384
dimples - if all the dimples were numbered, finding agree-
ment among 2 or more competitors as to which dimple is
"up" might be a real challenge.
Ed
"Mike Schneider" <super...@dot.com> wrote in message
news:supertitans-03...@c4-135.xtlab.com...
Gary Shannon
0..5? Call the red one the first digit and the blue one the second digit,
base 6. That gives you equal probability of any number from 00 to 55 base 6
= 0 to 35 base 10.
--gary
"Mike Schneider" <super...@dot.com> wrote in message
news:supertitans-03...@c4-135.xtlab.com...
Kevin J. Maroney
>I have a pair of 7-siders I picked up at a con. Two parallel pentagonal
>faces (numbered 6 and 7). 1 through 5 appear across the edges connecting
>the points of the two pentagons, so as to come out clearly on top when one
>of the 5 "sides" lands on the bottom. Rolls pretty fairly - the key was
>balancing the size of the sides with the faces.
I seem to recall someone posting (either here or over in the rpg
newsgroups) some testing he did that determined that the 7-sided dice
he had bought were, in fact, not particularly fair--that they tended
to land on their "end" faces more often than on the "side" faces.
--
Kevin J. Maroney | Unplugged Games | kmar...@ungames.com
Games are my entire waking life
Mike Schneider
<da...@desjardins.org> wrote:
> In this case, there's only one degree of freedom (the ratio of the
> length to the width). You could make a bunch of samples with different
> ratios, and try rolling them, until you come up with one that you are
> happy with. This is going to be more accurate than a simulation.
Given the computational power available today, it *should* be possible to
write a superior simulation which could tabulate dice thrown millions of
times from every conceivable angle. I wouldn't care to carve up a die,
roll it a thousand times, then carve a new with different characteristics,
roll it again a thousand times.... It's just too much work.
Mike Schneider
<fiz...@teleport.com> wrote:
> What's wrong with just tossing two different colored 6-sided dice numbered
> 0..5? Call the red one the first digit and the blue one the second digit,
> base 6. That gives you equal probability of any number from 00 to 55 base 6
> = 0 to 35 base 10.
Because we're *gamers*: We gotta have funky dice.
ccs-ed
36 outcomes? 1 - 2, 2 - 3's, . . . 6 - 7's, 5 - 8's, . . . & 1 -12.
We could also renumber just 1 die - faces of:
0, 6, 12, 18, 24 & 30.
The possible totals of the two die would then be:
1 to 36 - each with a equal probability.
Thanks Gary for showing me what was in front of me all the time.
Ed
Torben AEgidius Mogensen
>I have a pair of 7-siders I picked up at a con. Two parallel pentagonal
>faces (numbered 6 and 7). 1 through 5 appear across the edges connecting
>the points of the two pentagons, so as to come out clearly on top when one
>of the 5 "sides" lands on the bottom. Rolls pretty fairly - the key was
>balancing the size of the sides with the faces.
I would not allow these dice in any serious game (at least not if
fairness is of any consequence). I save a serious suspicion that the
chance of landing on one of the ends is greatly dependent on how you
roll the dice: It will be much higher if you rattle the die in a cup
and turn over the cup than if you roll the die along a table
(regardless of initial axis of rotation).
Likewise, I don't trust the "golfball" d100 that has been mentioned
earlier.
Torben
Torben AEgidius Mogensen
>Chessex has the 34-siders. They're basically two 17-sided pyramids,
>base-to-base.
If that was the case, it would land with an edge on top. To mae it
land with a face on top, you have to use kite-shaped faces like on a
d10.
Torben Mogensen (tor...@diku.dk)
Torben AEgidius Mogensen
>What's wrong with just tossing two different colored 6-sided dice numbered
>0..5? Call the red one the first digit and the blue one the second digit,
>base 6. That gives you equal probability of any number from 00 to 55 base 6
>= 0 to 35 base 10.
Or, correponding to rolling a d100 and a d10x10 and a d10 (where
d10x10 is a die with faces 00, 10, 20, ..., 90), you have one normal
d6 and a die with faces 0, 6, 12, 18, 24 and 30. Then you just have to
add the dice.
Torben Mogensen (tor...@diku.dk)
Torben AEgidius Mogensen
>Given the computational power available today, it *should* be possible to
>write a superior simulation which could tabulate dice thrown millions of
>times from every conceivable angle.
When rolling a die, the result is very sensitive to small influences,
so a simulation would have to take very small details into account to
be accurate enough to be useful. These details are the elasticity of
the die, elasticity of the surface, friction between surface and die,
air resistance etc.
After all, you you roll a die on a table and don't take friction into
account, it will just keep on rolling forever. And i you aren't
accurate, you won't be able to predict how far the die will roll.
A rolling die is a chaotic system (and deliberately so), which means
that it is practically impossible to predict the outcome even with
good simulations.
That said, you might not care about the accuracy of each individual
roll, as long as each end result comes up with the same probability.
But how can you convince anyone that this will be the case?
Besides, you are likely to miss something. Let us say you make a
perfect simulation of rolling a die with different spin and speed, on
different surfaces and so on and are satisfied with the fairness under
these circumstances (e.g. if the unfairness is smaller than the
variation you are likely to get due to imprecise manufacturing). Then
someone throws two dice in a single throw...
And lastly: Why bother making simulations to search for a fair d7,
when you can make a "virtual" d7 in at least two different, provably
fair ways:
1) Roll a d8 and reroll eights.
2) Make a fair d14 (possible, similar to d10) and label with 1-7
twice.
The first method requires a reroll one roll in 8, and possibly further
rerolls which adds up to 1 plus 1/7 rolls per final result. The
advantage is that teh average gamer will already have a d8.
The second method is, with the exception of a slight aesthetic
blemish, as good as a "real" d7 and, moreover, provably fair.
Torben Mogensen (tor...@diku.dk)
Torben AEgidius Mogensen
>Do you have a laptop and an Excel spreadsheet program?
>Just put this into a cell and press f9 every time:
>=TRUNC(RAND()*6)+1+TRUNC(RAND()*6)+1
Ouch! Don't trust the Excel random-number generator. At the very
least, use a cryptographically strong pseudo-random number generator
and seed it with external noise.
Torben Mogensen (tor...@diku.dk)
Bruno Wolff III
>
>(Shameless Plug Warning)
>
> If you have a Palm device, I invite you to check out DicePro. It
>will do what you want (along with a hundred other dice rolling duties
>you'll probably never need), and best of all, it's freeware!
And you have looked at the code or have done extensive testing to see
if the rolls it generates aren't biased?
There are a lot of very poor random number library routines and I tend
to be suspect of die rolling software unless I get some details on how
it generates the random rolls.
> http://www.geocities.com/TimesSquare/Realm/9565/
This link doesn't seem to be working right now.
Bruno Wolff III
>Tracy Johnson <t...@exis.net> writes:
>
>>=TRUNC(RAND()*6)+1+TRUNC(RAND()*6)+1
>
>Ouch! Don't trust the Excel random-number generator. At the very
>least, use a cryptographically strong pseudo-random number generator
>and seed it with external noise.
It also almost certainly has some (very small) bias just because 2^32
is not an even multiple of 6.
Often random number functions are pretty bad and it is probably worth
doing a little testing to make sure that the randomness is good enough
for your purposes before using one.
I have written a perl module that generates unbiased rolls while efficiently
using random bits. The random bits come from /dev/random which uses an
entropy pool and uses SHA-1 (a cryptographically strong hash function)
to extract random bits from that pool.
If anyone is interested in getting the code for the perl module or the
dice server (also written in perl), take a look at http://wolff.to/dice/ .
Mike Schneider
Mogensen) wrote:
> super...@dot.com (Mike Schneider) writes:
>
> >Given the computational power available today, it *should* be possible to
> >write a superior simulation which could tabulate dice thrown millions of
> >times from every conceivable angle.
>
> When rolling a die, the result is very sensitive to small influences,
> so a simulation would have to take very small details into account to
> be accurate enough to be useful. These details are the elasticity of
> the die, elasticity of the surface, friction between surface and die,
> air resistance etc.
If these variables are incorporated into the model (and, other than air
resistance, which should have negligible impact), it should then be a
simple matter of altering to see if different probabilities among facets
result. For instance, if a 12-sider has its pentagons trisected, two types
of facets result, with one type being twice as numerous as the other; if
one type of facet appears more frequently when rolled across felt or
rubber mats than when across a wooden table, then there's a problem.
> After all, you you roll a die on a table and don't take friction into
> account, it will just keep on rolling forever. And i you aren't
> accurate, you won't be able to predict how far the die will roll.
> A rolling die is a chaotic system (and deliberately so), which means
> that it is practically impossible to predict the outcome even with
> good simulations.
>
> That said, you might not care about the accuracy of each individual roll,
That's right: I don't, and it shouldn't matter because no individual
human rolls a die in exactly the same way in any event.
> as long as each end result comes up with the same probability.
> But how can you convince anyone that this will be the case?
>
> Besides, you are likely to miss something.
Probably, which is why there will be "Die Simulator 2.0", etc. A
simulation allows us to weed out bad ideas before we waste any time on
them in the *physical* world, which is exactly the same reason aircraft
are now designed and flown through cyberspace before the first piece of
aluminum is touched. If the physics of flight can be successfully
incorporated, I should think the matter of bouncing dice shouldn't be that
difficult.
David desJardins
tetrahedron, and replacing one of the flat faces with a convex face, so
that the die is unstable and won't ever come to rest on that face. Then
by symmetry it is equally likely to come to rest on each of the other
three faces.
One could do the same thing with two opposite faces of a cube, to make a
"4-sided die". Or with two opposite faces of a dodecahedron to make a
"10-sided die". In the latter case, if you collapse the two curved
surfaces to a point, then you get the "standard d10".
Can one use a similar combination of curved and flat surfaces to make
"fair dice" with other numbers of faces?
David desJardins
Kevin J. Maroney
>Can one use a similar combination of curved and flat surfaces to make
>"fair dice" with other numbers of faces?
Well, someone already suggested that you can make a die of an
arbitrary number of faces by taking an n-sided pipe (in the manner of
a standard pencil, which is a six-sided pipe) and rounding the ends so
that it can't stand on them.
David desJardins
> Well, someone already suggested that you can make a die of an
> arbitrary number of faces by taking an n-sided pipe (in the manner of
> a standard pencil, which is a six-sided pipe) and rounding the ends so
> that it can't stand on them.
Right. And similarly, an N-sided pyramid with a rounded base.
But, can one make an interesting "fair die" with more than two such
"rounded faces"?
David desJardins
Geenius at Wrok
Sure. You mentioned a d4 with one side rounded to make a d3. Wouldn't a
d6 with three adjacent sides rounded be effectively the same thing? (Of
course, you'd have to balance it properly, so that it didn't just wobble
around like a Weeble with the round side(s) down.)
--
"There is no greater joy for me than to find, on self-examination,
that I am true to myself." -- Mencius
=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+
Live with honor, endure with grace "I notice you have a cloud of doom.
Keith Ammann is gee...@enteract.com I must admit it makes you seem
www.enteract.com/~geenius * Lun Yu 2:24 dangerous and sexy."
Christopher Dearlove
<br...@cerberus.csd.uwm.edu> writes
>Often random number functions are pretty bad and it is probably worth
>doing a little testing to make sure that the randomness is good enough
>for your purposes before using one.
Well I'm also very suspicious of random number generators, but that's
in my professional capacity where I use millions of them and something
important might depend on the results.
Likewise if I owned a casino I'd be worried.
But for the purposes of this group does it really matter?
(As for the actual thread title, is there anyone who really can't add
up two figures? Actually I suspect many of us don't add them up any
more. I see the rectangular array of blobs and the X shape and say 11.
Consciously I probably don't even process the 6 and the 5 separately,
let alone actually add them up. Of course I may be wrong about what
I do, let alone anyone else.)
--
Christopher Dearlove
David desJardins
> Sure. You mentioned a d4 with one side rounded to make a d3. Wouldn't a
> d6 with three adjacent sides rounded be effectively the same thing?
No, that wouldn't be a fair die. Of the three remaining flat sides, the
one in the middle (adjacent to only two of the rounded sides) would
probably come up less often than the others. It's only provably "fair"
if each flat side is in an equivalent position relative to the rounded
sides.
David desJardins
Geenius at Wrok
I didn't mean three sides rounded in a row. I meant three sides in a
cluster, joined at one vertex. There couldn't be a "one in the
middle," because every flat side would be opposite one round side and
adjacent to the other two.
Clay Blankenship
You can get a "fair" die for more than one rounded face-e.g. the truncated
icosahedron (soccer ball) but rounding some faces is equivalent to
extending the other (flat) faces, so in that case you get back the regular
icosahdron.
Going back to an earlier part of the thread, I propose that even though the
truncated icosahedron (which can has 12 pentagons and 20 hexagons) does
not have the symmetry that all faces are equivalent (Sorry, I'm blanking on
the exact terminology), one could choose a ratio of hexagon size
to pentagon size (the degree of truncation) that would make each face
equally likely to come up, regardless of how it is thrown. (In contrast
to the seven-sider which is a right pentagonal prism, which could be
spun about its axis of rotational symmetry or not.)
--Clay
Clay Blankenship sno...@mindspring.com
"You keep using that word. I do not think that means what you think it means."
David desJardins
> I think he means the three rounded sides are mutually adjacent at a
> vertex.
OK, right. But in that case, it's the same as a 3-sided pyramid with a
rounded base. It works because it has rotational symmetry about an
axis. Are there "fair dice" with rounded faces that have a more
complicated symmetry group (and yet aren't just equivalent to a simpler
polyhedron)?
> You can get a "fair" die for more than one rounded face-e.g. the
> truncated icosahedron (soccer ball) but rounding some faces is
> equivalent to extending the other (flat) faces, so in that case you
> get back the regular icosahdron.
Right. Adding faces and then rounding them off doesn't count.
> I propose that even though the truncated icosahedron (which can has 12
> pentagons and 20 hexagons) does not have the symmetry that all faces
> are equivalent (Sorry, I'm blanking on the exact terminology), one
> could choose a ratio of hexagon size to pentagon size (the degree of
> truncation) that would make each face equally likely to come up,
> regardless of how it is thrown.
I really disagree with this. I think the probability of landing on a
hexagonal face will depend on whether the die is rolled with a high
horizontal velocity, or dropped with little horizontal velocity, or
launched with a lot of spin, or without spin, etc.
David desJardins
P.S. I think there's no standard mathematical language for "the symmetry
group is transitive on the faces", as opposed to the vertices. I guess
you could say that the dual polyhedron is uniform.
Gary Shannon
> In article <%GYa7.3872$Ll5.9...@nntp1.onemain.com>, "Gary Shannon"
> <fiz...@teleport.com> wrote:
>
> > What's wrong with just tossing two different colored 6-sided dice
numbered
> > 0..5? Call the red one the first digit and the blue one the second
digit,
> > base 6. That gives you equal probability of any number from 00 to 55
base 6
> > = 0 to 35 base 10.
>
>
> Because we're *gamers*: We gotta have funky dice.
>
Hmmm. I would think that funky mathematical recipes would be just as fun.
After all, we already use funky tables for combat outcomes and wandering
monsters. As long as it's funky who cares whether it's a die or a table?
:-)
--gary
The Doctor
>
> What's wrong with just tossing two different colored 6-sided dice numbered
> 0..5? Call the red one the first digit and the blue one the second digit,
> base 6. That gives you equal probability of any number from 00 to 55 base 6
> = 0 to 35 base 10.
It's kindof hard to read 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7,
7, 7, 7, 7, 7, 7 etcetera in that.
You'd have to memorize 00 to be 2, 01 to be 3, 55 to be 12, 43 to be
erm.. 9? etc.
one die that simply shows the right number would be much easier then
some new complicated translation that will replace the adding up.
//Doc.
Mike Schneider
<da...@desjardins.org> wrote:
> It seems to me that one could make a fair "3-sided die" by taking a
> tetrahedron, and replacing one of the flat faces with a convex face, so
> that the die is unstable and won't ever come to rest on that face. Then
> by symmetry it is equally likely to come to rest on each of the other
> three faces.
>
> One could do the same thing with two opposite faces of a cube, to make a
> "4-sided die". Or with two opposite faces of a dodecahedron to make a
> "10-sided die". In the latter case, if you collapse the two curved
> surfaces to a point, then you get the "standard d10".
The dice you describe above would have equal-probability characteristics
since each flat facet has an equal number and arrangement of adjoining
facets and convex faces. Larger dice would need to preserve this
characteristic: If we assume for the moment that 38-sided "perfect"
platonic solid existed, merely convexing two of its facets would be
insufficient to yield a 36-sider since the probability of landing on
facets near a convex faces would differ from elsewhere on the die.
(BTW, all dice but the tetrahedron 4-sider can be convexed easily by
imagining the die as cut from a sphere. The tetrahedron would need an
internal weight near its "peak", because otherwise its low center of
gravity would still enable it to occasionally come to rest on the convex
face)
Torben AEgidius Mogensen
Yes. You can make an N-sided prism and taper the ends to a point. A
3-sided die made like this will look like a straight banana. This can
be done for any N, but will be impractical for big N's. Klaus' webpage
lists the possible curved-face fair dice as well as the polyhedrical
fair dice.
I think it will be more practical to make odd-sided dice as dice with
2N faces numbered from 1-N twice. But even then, it becomes
impractical for big N's.
Someone might be tempted to make, e.g., a d11 by taking a d12 and
round one face. But that will not make the die fair, as the 5 faces
that neighbour the rounded face will be more likely than the 6
opposite faces.
Torben Mogensen (tor...@diku.dk)
David desJardins
> Yes. You can make an N-sided prism and taper the ends to a point. A
> 3-sided die made like this will look like a straight banana. This can
> be done for any N, but will be impractical for big N's. Klaus' webpage
> lists the possible curved-face fair dice as well as the polyhedrical
> fair dice.
The web page only includes those dice for which all faces are identical.
It doesn't attempt to analyze dice which have two types of faces: flat
or curved faces, all of which have the same probability; and curved
faces which are unstable (so have probability zero).
In principle, a die could even have curved unstable regions that are not
faces (for example, which are not convex). Think, for example, of a
sphere with six flat faces cut out to form the six sides of a cube.
Then all of the faces have the same probability (although it's not clear
that the curved surface can be made to have no stable points). Of
course, this is just equivalent to a cubical die, but it's not clear (to
me) that all "fair dice" with curved and flat surfaces are so trivial.
David desJardins
Torben AEgidius Mogensen
>But, can one make an interesting "fair die" with more than two such
>"rounded faces"?
Depends on what you mean by "interesting". You can make a d12 by
taking a d20 and round off two opposite ends and the three faces that
meet these ends. Similarly, you can make a d12 by taking a d20 and
round off the 10 faces that touch two opposing vertices. However, you
will find that these are topologically identical to, respectively, a
d12 make with two 6-sided pyramids and the standard d10.
Torben
Mike Schneider
Blankenship<sno...@mindspring.com> wrote:
> Going back to an earlier part of the thread, I propose that even though the
> truncated icosahedron (which can has 12 pentagons and 20 hexagons) does
> not have the symmetry that all faces are equivalent (Sorry, I'm blanking on
> the exact terminology), one could choose a ratio of hexagon size
> to pentagon size (the degree of truncation) that would make each face
> equally likely to come up, regardless of how it is thrown.
Precisely. And how is that ratio determined?
Don Woods
> In article <nLFb7.956$NJ6....@www.newsranger.com>, Clay
> Blankenship<sno...@mindspring.com> wrote:
> > one could choose a ratio of hexagon size
> > to pentagon size (the degree of truncation) that would make each face
> > equally likely to come up, regardless of how it is thrown.
>
> Precisely. And how is that ratio determined?
That's one level of question, and one possible answer is that
we don't know how to determine it, but that doesn't mean there
is no such value. I.e., as you vary the ratio from one extreme
to another, there must be some point at which the die is "fair",
even if we don't know how to determine when that point is reached.
However, elsewhere in this thread, David desJardins raised the
next level of question, namely, is it in fact the case that there
must be some ratio that makes the die fair? Or is it possible
that the ratio that makes for a fair die if the die is rolled
in a certain way (e.g., starting with a hexagonal face at the
top and rolling in a certain direction with a certain momentum)
is not fair if the die is rolled in some other manner?
-- Don.
Mike Schneider
Mogensen) wrote:
> Yes. You can make an N-sided prism and taper the ends to a point. A
> 3-sided die made like this will look like a straight banana. This can
> be done for any N, but will be impractical for big N's. Klaus' webpage
> lists the possible curved-face fair dice as well as the polyhedrical
> fair dice.
URL?
> I think it will be more practical to make odd-sided dice as dice with
> 2N faces numbered from 1-N twice. But even then, it becomes
> impractical for big N's.
>
> Someone might be tempted to make, e.g., a d11 by taking a d12 and
> round one face. But that will not make the die fair, as the 5 faces
> that neighbour the rounded face will be more likely than the 6
> opposite faces.
They should be *less* likely to appear, since it is the "last bounce"
which causes the die to finally stop rolling. If a die would have stop
after a "last bounce" on any given face, then removing a "ridge" by
rounding the die means it will maintain momentum over to the next face,
bonk the ridge on the far side, and *then* come to a stop *two* faces away
from the round.
Torben AEgidius Mogensen
>In article <9ko905$d...@grimer.diku.dk>, tor...@diku.dk (Torben AEgidius
>Mogensen) wrote:
>> Yes. You can make an N-sided prism and taper the ends to a point. A
>> 3-sided die made like this will look like a straight banana. This can
>> be done for any N, but will be impractical for big N's. Klaus' webpage
>> lists the possible curved-face fair dice as well as the polyhedrical
>> fair dice.
>URL?
It has been mentioned several times during this thread, but here it is
again: http://hjem.get2net.dk/Klaudius/Dice.htm.
Torben
Bruno Wolff III
>
>But for the purposes of this group does it really matter?
Some random number generators are bad enough that I would be concerned about
using them in board and/or RP games. In particular I would be worried about
short cycles of high/low rolls or dependencies bewtween consectutive rolls.
In many games the players can manipulate the order of rolls and could take
advantage of the patterns listed above.
Torben AEgidius Mogensen
I once wrote a simulator of battles for Britannia, with the purpose of
estimating the chances of each possible final outcome (I was too lazy
to work it out precisely). Some of the results seemed very odd, so I
changed the random number generator and redid the simulation, which
gave the expected result. The number of rolls was large enough that
the difference wasn't likely to be caused by statistical variation.
The moral is that even for games, a bad PRNG isn't acceptable.
Torben Mogensen (tor...@diku.dk)
Christopher Dearlove
<tor...@diku.dk> writes
>I once wrote a simulator of battles for Britannia, with the purpose of
>estimating the chances of each possible final outcome
Now that's a simulation, not just a single game. Definitely a need
for a decent RNG in this case.
--
Christopher Dearlove
Christopher Dearlove
<de...@math.berkeley.edu> writes
>OK, right. But in that case, it's the same as a 3-sided pyramid with a
>rounded base. It works because it has rotational symmetry about an
>axis. Are there "fair dice" with rounded faces that have a more
>complicated symmetry group (and yet aren't just equivalent to a simpler
>polyhedron)?
Beyond the set of Platonic solids (all faces equivalent, all vertices
equivalent) there is the set of Archimedean solids which may have more
than one type of face, but all faces of each type is equivalent and
all vertices of each type are equivalent. (I think I have the definition
correct). You could then round off all the faces of all but one type.
(Although not all such cases would work.) Many are not convex, so you
have to discard them, and many are truncated Platonic solids - which if
you pick the truncation faces won't help.
The only presentation of these solids I've seen is in the Science
Museum (*) in Kensington, London. There are models of all of them
shown (**). Unfortunately I won't be there again for a bit (it's
due to be free again soon though!) to verify.
(*) Well that's what it used to be. It's something like the
Museum of Science and Industry now (which is actually a
better name, but not perfect).
(**) Except only representatives of the two infinite sets of
Archimedean solids - left as an exercise to the reader :)
--
Christopher Dearlove
Paul Davidson
> more. I see the rectangular array of blobs and the X shape and say 11.
> Consciously I probably don't even process the 6 and the 5 separately,
> let alone actually add them up. Of course I may be wrong about what
> I do, let alone anyone else.)
> --
> Christopher Dearlove
You're correct - most people just look at the dot pattern, rather than count
each dot.
In fact, I believe that a person's capacity to "know" how many
dots/apples/etc. are in front of them without having to count is actually a
measure of some aspect of one's IQ.
Paul Davidson
Man In The Doorway
> (Shameless Plug Warning)
>
> If you have a Palm device, I invite you to check out DicePro. It
> will do what you want (along with a hundred other dice rolling duties
> you'll probably never need), and best of all, it's freeware!
>
> http://www.geocities.com/TimesSquare/Realm/9565/
Just went to that web site, following the hyperlink.
I like it.
And if it's free for the downloading you're clear of the plug label,
so relaxxxx
Man In The Doorway
>
> Ouch! Don't trust the Excel random-number generator. At the very
> least, use a cryptographically strong pseudo-random number generator
> and seed it with external noise.
I actually understand what this means.
*arms hugging me*
I love myself.
Man In The Doorway
> I have written a perl module that generates unbiased rolls while efficiently
> using random bits. The random bits come from /dev/random which uses an
> entropy pool and uses SHA-1 (a cryptographically strong hash function)
> to extract random bits from that pool.
How did you go about programming this?
I used to ... know a little code. So, how? be detailed
Mike Schneider
Imagine your "soccer ball" dice with hexagons and pentagons. No matter how
a throw is initiated, the ensuing chaos of bouncing (and turning in air)
renders orientation thoroughly random very quickly. This means that speed
and direction of the initial throw can be completely ignored, provided we
demand it is sufficient to get the die to bounce several times (a "bounce"
meaning each facet-turn of the die while in contact with the table -- a
roll is a series of small bounces), and focus exclusively on the mechanics
of the "final bouncing" before the die comes to rest. This should vastly
reduce the amount of complexity necessary for a simulator.
Mike Schneider
Davidson" <tin...@theforce.net> wrote:
I disagree. There may be "some aspect", but IMO, an *unconscious* thought
process which arrives at some level of accuracy is not "intelligence". An
aborigine hunter who can skewer a running wild pig at fifty yards with a
thrown spear without *thinking* through the physics of the throw is not
"intelligent" any more than Duston Hoffman's "Rainman" character is
intelligent for his ability to instantly count a jumbled pile of dropped
toothpicks.
John R. Cooper
III) wrote:
>On Fri, 03 Aug 2001 14:33:12 -0700, John R. Cooper <jo...@jrcooperREMOVE.com> wrote:
>>
>>(Shameless Plug Warning)
>>
>> If you have a Palm device, I invite you to check out DicePro. It
>>will do what you want (along with a hundred other dice rolling duties
>>you'll probably never need), and best of all, it's freeware!
>
>And you have looked at the code or have done extensive testing to see
>if the rolls it generates aren't biased?
No, I do not have access to the PalmOS SysRandom() function source
code. If anyone is sufficiently motivated in doing some tests of
DicePro's randomization, I'd be curious to see the results.
>There are a lot of very poor random number library routines and I tend
>to be suspect of die rolling software unless I get some details on how
>it generates the random rolls.
Fair enough. I get the impression, though, that hardly anyone is
too concerned with this issue. In the (nearly) four years DicePro has
been knocking around, I've received only two or three e-mails asking
me about it. *Shrug*
Cheers,
- John
George W. Harris
<de...@math.berkeley.edu> wrote:
:Kevin J. Maroney <kmar...@ungames.com> writes:
:> Well, someone already suggested that you can make a die of an
:> arbitrary number of faces by taking an n-sided pipe (in the manner of
:> a standard pencil, which is a six-sided pipe) and rounding the ends so
:> that it can't stand on them.
:
:Right. And similarly, an N-sided pyramid with a rounded base.
:
:But, can one make an interesting "fair die" with more than two such
:"rounded faces"?
One could turn an eight-sided die into a
four-sided die by doing such to four fo the faces; two
opposite faces on each vertex would be rounded (on
the eight-sided dice I have this would be either all the
even faces or all the odd faces). One could turn a
20-sided die into a 12-sided die by rounding 8 faces
(choose a face; then choose three faces which each
share a different, single vertex with that face, and no
two of which share borders with another common face.
Round all four of those and their opposites), leaving six
pairs of adjacent faces whose common edges are
oriented as the faces of a cube (or vertices of an
octahedron), or round those 12 sides to get an 8-sided
die. Converting a d12 to a d6 involves either rounded
the six faces adjacent to two opposite vertices, or the
six faces *not* adjacent to two opposite vertices. All of
these possibilities are less interesting as there are other,
simpler dice with these numbers of faces, just as no one
will ever produce any of the variant fair d12s
: David desJardins
George W. Harris For actual email address, replace each 'u' with an 'i'
Tracy Johnson
>
> In fact, I believe that a person's capacity to "know" how many
> dots/apples/etc. are in front of them without having to count is actually a
> measure of some aspect of one's IQ.
I find this strikingly similar to cashiers who can still provide
the correct change prior to the register's display telling them
how much. For example, did you ever tender ten dollars for a
charge of nine dollars and then watch the cashier 'wait' for
the display to tell them to provide one dollar in change?
I seem to notice the ability to provide correct change manually
has declined over the years. Is it because retailers are hiring
more stupid people or is it the 'dumbing down' of America?
--
BT
NNNN
Tracy Johnson
Justin Thyme Productions
Sponsors Free Multiuser Wargaming on the WEB at:
http://hp3000.empireclassic.com/
Eric Haas
<de...@math.berkeley.edu> wrote:
>It seems to me that one could make a fair "3-sided die" by taking a
>tetrahedron, and replacing one of the flat faces with a convex face, so
>that the die is unstable and won't ever come to rest on that face. Then
>by symmetry it is equally likely to come to rest on each of the other
>three faces.
>
>One could do the same thing with two opposite faces of a cube, to make a
>"4-sided die". Or with two opposite faces of a dodecahedron to make a
>"10-sided die". In the latter case, if you collapse the two curved
>surfaces to a point, then you get the "standard d10".
I have a 4-sided die like you describe. Although, actually, they've
made the square sides into rectangles but, the idea is the same. The
top and bottoms are pyramids so it can only land on it's sides.
I wish I could find more dice like that.
Torben AEgidius Mogensen
>[skip] All of
>these possibilities are less interesting as there are other,
>simpler dice with these numbers of faces, just as no one
>will ever produce any of the variant fair d12s
Shame, really. I would love to have a rhombic dodecahedron d12. :-)
Torben Mogensen (tor...@diku.dk)
Bruno Wolff III
>br...@cerberus.csd.uwm.edu (Bruno Wolff III) wrote in message news:<slrn9mtjim...@cerberus.csd.uwm.edu>...
>
>
>I used to ... know a little code. So, how? be detailed
I am not sure what you are asking. The source code is available from the web
site (http://wolff.to/dice/) mentioned in the post you responded to.
A brief description of the strategy follows:
The main assumption is that using random bits is the only significant cost
to the calculation (when using /dev/random, I think this is a reasonable
assumption).
Two numbers are maintained while generating a random integer in a specific
range (0 to N-1). The first number is the current range, and the second
is the current random number.
The starting current range is 1.
The starting current random number is 0.
If the current range is < N then a 0 bit is added on the end of the
current range and a random bit from /dev/random is added to the end
of the current random number. And then the check is repeated.
If the current range >= N and the current random number < N,
then the current random number is returned.
If both the current range and the current random number are >= N, then
N is subtracted from both and the check is repeated.
Bruno Wolff III
>
>You're correct - most people just look at the dot pattern, rather than count
>each dot.
It is worse than that, it actually takes me longer to read dice (at least
6 siders) with numbers on them, than ones with dots.
Ian Noble
<gha...@mundsprung.com> wrote:
>On Fri, 03 Aug 2001 23:17:47 GMT, "Jason E. Schaff"
><jason...@home.com> wrote:
>
>:I would think that there must be some geometry that would produce a "fair"
>:36-sided die though. I have actually seen a 100-sided die for sale once:
>:the thing was _huge_, about the size of a plum.
>
> The only geometry that produces a "fair"
>36-sided die would be one which consists of two
>18-sided (where the ones at Giza are considered
>4-sided) pyramids base-to-base.
(OK - probably this ought to be for rec.mathwhatever, but it's too
long since I did any of this to be able to frame it in those terms -
and in any case, one of these days I'll get around to actually
exploring this area further...)
It seems to me that a fair die needn't have all its faces the same
shape, if by fair you simply mean "no discernable statistical bias".
For example, there are at least two possibilities for dice having
faces of two different shapes that *might* still, statistically, be
fair. I'd be interested to know if anyone knows of anyone
investigating either of them.
To explain one of them (a 14-sided die): Take a cube and trim a
*small* portion off every corner, keeping the result as symmetric as
you can, to form 8 small equilateral triangular faces. The resulting
solid has 14 faces - the 8 small triangles and 6 octagons. If you use
it as a die, it will heavily favour the much larger octagons - the
chances of ot landing on a triangle are small. Keep trimming the
corners, making the triangles larger and larger, and eventually their
corners will meet. At this point the shape (a cuboctahedron? I
forget) has 8 triangular faces and 6 square ones.
Now start the same sort of corner-trimming exercise on an octahendron.
This time the small faces are square - and initially, again, there's
very little chance of the die landing on them rather than one of the
much larger (hexagonal this time) faces. But keep going, and
eventually the corners of the growing squares meet - producing the
*same* 8 triangle/6 square shape. In other words, we can produce a
smooth continuum of 14-sided shapes between a cube and an octahedron,
where at one end of the spectrum chance heavily favours the faces that
become those of the cube, whilst at the other end it heavily favours
those that become the faces of the octahedron. Intuitively, somewhere
in between (skewed, if the area of the faces is any guide, towards the
octahedron end of the scale) perhaps lies a "fair" 14-sided die.
The question is, of course, whether such hybrid shapes actually have a
single probability distribution irrespective of throwing conditions,
or whether their behaviour is altered by, say, the velocity at which
they're thrown (and if so, to what degree - as a gamer I could accept
a known, small variation in behaviour - whilst a large variation could
be an interesting property in itself).
Clearly, if it works at all, the same approach could generate a "fair"
32-sided die lying somewhere between the dodecahendron and the
icosahedron. It won't work to generate anything new starting with a
tetrahedron - you simply trim down to a regular octahedron (clearly
fair, but a shape we already know about), then keep slicing bits off
its faces until you finally arrive at another tetrahedron.
Cheers - Ian Noble.
Bruno Wolff III
I don't think this is true. I don't think the spin of the die is going to
be properly randomized by a couple of bounces. I also don't think you can
just look at the final bounce, because your assumption about having a
chance to randomize orientations isn't going to apply to the time between
the next to last and last bounces. There are going to be preferred
orientations coming into the last bounce because there will be preferred
orientations coming out the the second to last bounce.
Ian Noble
(Darn - just caught up with this part of the thread. Sorry for
dredging up something that had already been pointed out.)
Cheers - Ian Noble
Torben AEgidius Mogensen
>A brief description of the strategy follows:
>The main assumption is that using random bits is the only significant cost
>to the calculation (when using /dev/random, I think this is a reasonable
>assumption).
>Two numbers are maintained while generating a random integer in a specific
>range (0 to N-1). The first number is the current range, and the second
>is the current random number.
>The starting current range is 1.
>The starting current random number is 0.
>If the current range is < N then a 0 bit is added on the end of the
>current range and a random bit from /dev/random is added to the end
>of the current random number. And then the check is repeated.
>If the current range >= N and the current random number < N,
>then the current random number is returned.
>If both the current range and the current random number are >= N, then
>N is subtracted from both and the check is repeated.
When I saw this, I said "surely, the third step must be wrong". But I
failed to find a case where it gives a bad distribution, so it is
probably O.K. But I don't think it is optimal in the number of
required random bits. It is probably better than starting over if the
current value >=N, though.
For optimality, you would need arithmetic encoding or something
similar, which requires you to store some state between generating
numbers. Intuitively, you need a fractional number of bits to generate
a number in a range that is not a power of 2, so you would want to
store the unused bit-fraction for the next number. It is, obviously,
nonsense to store a fraction of a bit, but you can get the same effect
if you have a (theoretically) infinite sequence of random bits.
Torben Mogensen (tor...@diku.dk)
Bruno Wolff III
>br...@cerberus.csd.uwm.edu (Bruno Wolff III) writes:
>
>When I saw this, I said "surely, the third step must be wrong". But I
>failed to find a case where it gives a bad distribution, so it is
>probably O.K. But I don't think it is optimal in the number of
>required random bits. It is probably better than starting over if the
>current value >=N, though.
The third step just handles the remainder when there aren't a number of
states divisible by N.
It is clearly not best to just start over. This throws away information,
and clearly is not optimal.
I believe my method is the optimal solution for generating a uniform
random distribution over the range 0 to N-1 giving a source of random
bits.
>For optimality, you would need arithmetic encoding or something
>similar, which requires you to store some state between generating
>numbers. Intuitively, you need a fractional number of bits to generate
>a number in a range that is not a power of 2, so you would want to
>store the unused bit-fraction for the next number. It is, obviously,
>nonsense to store a fraction of a bit, but you can get the same effect
>if you have a (theoretically) infinite sequence of random bits.
The method you are suggesting doesn't seem to say anything useful
(as fractional bits don't exist as such) and is not an optimal way
to do things. If you have a series of random numbers that you want to
generate using the least amount of entropy, then you want to do them
all at once. You treat the whole combination or rolls as one event
and use the normal method to pick one outcome of this combined event
at random.
If you look at the perl module I wrote, you will find code that takes
some advantage of that. One routine is used for rolling a bunch of
the same sided dice (a common case in practice). It bunches as many
rolls together that can have a total number of possiblities less than
a number that is safe for doing operations on using normal operators.
Christopher Dearlove
<br...@cerberus.csd.uwm.edu> writes
>I believe my method is the optimal solution for generating a uniform
>random distribution over the range 0 to N-1 giving a source of random
>bits.
Let me just clarify my understanding
For any given N any method such as yours will consume, a mean number -
let's call it a(N) - of bits. You are claiming your algorithm gives
minimal a(N) for all N.
Actually I've just realised (after typing most of what is below, which
I will now edit) that there is still an ambiguity in this claim. Are
you claiming:
- A minimal mean a(N) for a single number, or
- A minimal mean a(N) for a sequence of numbers (probably in the
limit as the sequence length tends to infinity)?
I believe your claim for the first of these two (but haven't tried
to prove it). I am less sure of the second claim - which is probably
that of real interest. Hereafter I'll consider that case.
If you happen to have results for a(N) (experimental will do initially)
this would be very useful to check my thought below. (I don't have a
source of bits which I trust to hand, and I'm feeling too lazy to
do the calculation of interest below, considering all the special
cases of intermediate range values.)
Anyway here is an argument that may be of some interest.
Consider N=3. This gives an a(3) which is easy to calculate by
a(3) = (3/4) x 2 + (1/4) x (2 + a(3)), hence giving a(3) = 8/3.
[This is because N=3 gains nothing from the interesting stage.]
Now consider N=9. I believe that if your algorithm is ideal then
a(9) must be 16/3. The question is, is it?
My argument goes that suppose a(9) is more than 16/3, then
instead of using your algorithm for N=9, use it twice for N=3
and join the two numbers together. Now suppose instead that
a(9) is less than 16/3, then instead of using your algorithm for
N=3, use it for N=9 and split into two numbers. Hence if your
algorithm is optimal for N=3 and N=9 then a(9) must equal 16/3.
Of course if either of these alternatives applies there may be yet
further tricks; and generalising the trick to any N would be
horrible, and you may well be better off with your elegant algorithm.
I'm just considering the optimality claim however.
--
Christopher Dearlove
Jordan Wolbrum
"John R. Cooper" wrote:
> No, I do not have access to the PalmOS SysRandom() function source
> code. If anyone is sufficiently motivated in doing some tests of
> DicePro's randomization, I'd be curious to see the results.
>
Someone posted a website a while ago that lets you enter totals and then it will suggest
how fair the dice are.
Since I have neither the URL or a Palm device, I can't be much more help than to suggest
Google/Deja.
-JW
Mike Schneider
br...@cerberus.csd.uwm.edu (Bruno Wolff III) wrote:
> On Tue, 07 Aug 2001 16:05:46 -0500, Mike Schneider <super...@dot.com>
wrote:
> >In article <7witg0b...@ca.icynic.com>, Don Woods <d...@iCynic.com> wrote:
> >
> >Imagine your "soccer ball" dice with hexagons and pentagons. No matter how
> >a throw is initiated, the ensuing chaos of bouncing (and turning in air)
> >renders orientation thoroughly random very quickly. This means that speed
> >and direction of the initial throw can be completely ignored, provided we
> >demand it is sufficient to get the die to bounce several times (a "bounce"
> >meaning each facet-turn of the die while in contact with the table -- a
> >roll is a series of small bounces), and focus exclusively on the mechanics
> >of the "final bouncing" before the die comes to rest. This should vastly
> >reduce the amount of complexity necessary for a simulator.
>
> I don't think this is true. I don't think the spin of the die is going to
> be properly randomized by a couple of bounces.
*One* bounce of any sufficient force (a one-foot throw) should suffice if
it randomly alters the spin-axis of the dice by a radial degree equal to
the ragial angle between one facet center and its edge (i.e., not much).
Not to mention that the initial spin of the die as it leaves the thrower's
hand is also random.
> I also don't think you can
> just look at the final bounce, because your assumption about having a
> chance to randomize orientations isn't going to apply to the time between
> the next to last and last bounces. There are going to be preferred
> orientations coming into the last bounce because there will be preferred
> orientations coming out the the second to last bounce.
We're not interested in where the "final bounce" came from -- only where
it's going.
Bruno Wolff III
>
>Let me just clarify my understanding
>
>For any given N any method such as yours will consume, a mean number -
>let's call it a(N) - of bits. You are claiming your algorithm gives
>minimal a(N) for all N.
>
>Actually I've just realised (after typing most of what is below, which
>I will now edit) that there is still an ambiguity in this claim. Are
>you claiming:
>
>- A minimal mean a(N) for a single number, or
>- A minimal mean a(N) for a sequence of numbers (probably in the
> limit as the sequence length tends to infinity)?
It provides the minimal mean for a set of rolls where the range
of each roll is known before any rolls are made. (This allows
the roll to be done as one super roll.)
>I believe your claim for the first of these two (but haven't tried
>to prove it). I am less sure of the second claim - which is probably
>that of real interest. Hereafter I'll consider that case.
>
>If you happen to have results for a(N) (experimental will do initially)
>this would be very useful to check my thought below. (I don't have a
>source of bits which I trust to hand, and I'm feeling too lazy to
>do the calculation of interest below, considering all the special
>cases of intermediate range values.)
What you had below appeared to be correct and shows that it pays to
combine rolls where you can. I think to be able to do that usefully
you need to know what the future roll requests are going to be.
I actually do that optimization in my code for cases where a group
of rolls in the same range are requested at once. This is bounded
by the word size used for the calculations as a trade off for
simplicity and speed in the common cases.
Guus
this page i will forbid my kids ever to roll two dices at the same
time, preventing them comming with the solution these students found
(http://www.ouc.bc.ca/phys/dbmu/dice/phapr01b.jpg) for simply rolling
two dices.
On 3 Aug 2001 13:59:00 -0700, shu...@yahoo.com (Justin Green) wrote:
>"Arthur Gibbs" <nos...@aol.com> wrote in message news:<3GAa7.84260$TM5.10...@typhoon.southeast.rr.com>...
>> Anyone know of a place I can buy one 36 sided die? One that has the numbers
>> that would show up as if you rolled two six sided dice? I know this sounds
>> lazy.
>
>I can't help you, but here's a machine that will roll two dice every
>2.5 secs (34560 rolls per day...or approx. 450 games of Settlers). It
>takes a picture of the dice after each roll and adds the numbers
>together for you, so you don't have to worry about THAT. You'd
>probably have to hire a Canadian physics grad student or two to work
>and maintain it...but they don't cost much. ;-)
>
>http://www.ouc.bc.ca/phys/dbmu/dice/
George W. Harris
Mogensen) wrote:
As would I. Or any of the possible d24s (random hour of the
day in a single die!).
: Torben Mogensen (tor...@diku.dk)
--
"Intelligence is too complex to capture in a single number." -Alfred Binet
George W. Harris
wrote:
:Paul Davidson wrote:
:>
:> In fact, I believe that a person's capacity to "know" how many
:> dots/apples/etc. are in front of them without having to count is actually a
:> measure of some aspect of one's IQ.
:
:I find this strikingly similar to cashiers who can still provide
:the correct change prior to the register's display telling them
:how much. For example, did you ever tender ten dollars for a
:charge of nine dollars and then watch the cashier 'wait' for
:the display to tell them to provide one dollar in change?
:
:I seem to notice the ability to provide correct change manually
:has declined over the years. Is it because retailers are hiring
:more stupid people or is it the 'dumbing down' of America?
I have discovered that a firghtening number of
Massachusetts sixth graders think there are sixty cents
in a dollar.
--
When Ramanujan was my age, he had been dead for seven years. -after Tom Lehrer
Gary Shannon
Select the range, push the button, read the results.
Somebody ask Intel if they'll design a custom chip for it.
--gary
"George W. Harris" <gha...@mundsprung.com> wrote in message
news:cul3ntc01dbrs7n3v...@4ax.com...
Scott Alan Woodard
news:nUnc7.3153$7g6.5...@nntp1.onemain.com...
> Sounds like what every gamer really needs is a programmable electronic
die.
> Select the range, push the button, read the results.
>
> Somebody ask Intel if they'll design a custom chip for it.
>
> --gary
>
it was called a "Dragonbone" or something like that and it was a handheld
device that allowed you to select several different types of "die" as well
as different quantities. I don't think people actually really WANTED an
alternative to mounds of polyhedrals, though...
Scott Alan Woodard
ogm...@earthlink.net