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DODGY
DICE EXPLAINED
loaded high
and low thrower twice
dice regular how
to tell them apart Vincent's
Formula
HOLD
THE CURSOR OVER A DIE TO SEE THE OTHER SIDE
 Loaded
Dice:
These contain a hidden weight at the opposite side to the number
required. When rolled, the die is likely to settle with the weight
at the lowest point, therefore the desired number will be uppermost.
Success rate depends on how hard you throw it and the friction of
the rolling surface. Judge the strength of throw to avoid the die
being abruptly stopped by an obstacle.
Avoid playing on a smooth, shiny surface where the die may slide
rather than roll. These are heavily loaded dice. Once you are familiar
with their behaviour, it is possible to throw the desired number
up to 90% of the time. For more information on the mathematics of
loading dice than you could possibly want, see Vincent's
formula at the bottom of this page.
  High
and Low Thrower:
The high thrower always throws a high number because it only has
4s, 5s and 6s on it. The duplicated numbers are on opposite sides
to each other, and since only 3 sides of a cube can be seen at any
one time, the die will appear normal whichever angle it is viewed
from. The Low Thrower is its complement. This one always throws
a low number because it only has 1s, 2s and 3s on it.
The person throwing the high thrower will always beat someone using
the low thrower. It's not difficult to work out why, but you will
be amazed how seldom the deception is spotted. Most dodgy dice combinations
have an element of probability, but this pair guarantees 100% success.
 Twice
Dice:
Each of these dice has had one number on it replaced by another,
for example the double fiver has an extra 5 instead of a 2. The
extra number is always on the opposite face to its double, so they
are never visible at the same time. Only a close examination will
give the game away.
This small change makes a huge difference. Obviously, these dice
can double your chances of throwing any particular number, but more
than that, you can quite precisely control the advantage you have
over your opponent in some dice games. Supposing the object of the
game is to move your counters round a board as fast as possible,
according
to the throws of the dice. Say your opponent has an ordinary die
with 21 spots and you use the double fours which has 22. You have
a 4.76% advantage over him. If you use the regular die and give
him the double twoer (18 spots), it goes up to 16.66%. Want a bigger
advantage? Throw for throw, the double sixer will lose to the double
oner only one throw in six. How about reducing his chances to zero
by letting him use the double oner for a game where you need a six
to start?!
 Regular:
In some ways, this is the most important one of all, a normal honest
die. The sort of die you can leave lying about, let other people
use, because it's nothing special, just a regular die. It just happens
to look identical to the Dodgy Dice in your pocket.... Lull your
victim into a false sense of security, then strike like a cobra.
How
you can tell them apart:
Don't worry if your dodgy dice get mixed up. Wobble loaded dice
in the palm of your hand to distinguish them from regular dice.
If it's loaded, one face will consistently remain uppermost. Twice
dice and high and low throwers can be recognised by examining the
number of spots.
Vincent's
Formula:
My contribution to the noble art of dice-loading
You probably imagine the life of a dodgy dice manufacturer is filled
with adventures in Las Vegas and fighting off backgammon groupies.
Whilst I will neither confirm nor deny that impression, I would
like you to consider the other side of the job, the research, the
experimentation, the striving for perfection.....
It was January in the Highlands, a time for sitting by the fire
and thinking. I'd been intermittently working on the design for
a new type of loaded die, one I hope to introduce to this website
soon. Design constraints meant that I wouldn't be able to use as
dense a loading material as I usually do (lead), so it was necessary
to optimise the other aspects affecting the bias : thickness of
walls, lightness of walls, and depth of loading. I'd made good progress
on the first two, and now I was considering the third. Common sense
and a bit of experimentation would get me pretty close to the ideal
depth, but the difference might well mean one or two percent less
bias, and it was precisely this untapped potential that I was looking
for. So I dug out an old notebook and a pencil and threw another
log on the fire. And I thought to myself....
Imagine a hollow die. It's centre of gravity (CoG) is at the centre
of its volume. Add a little weighting material to the bottom of
the hollow and the CoG lowers. As you add more material, the CoG
descends until you overfill it and the CoG begins to rise. The optimum
depth is when the CoG coincides with the top surface of the loading
material. The actual depth will depend upon the size of the die,
the thickness of the walls, and the relative densities of
the dice material and the loading material. I wondered if this was
enough information to create a universal dice loading formula, one
which would give the optimum loading depth for a die of any given
size, wall thickness and materials.
It seemed to me that the key was the fact that at this optimum depth,
the distance from the centre, of both the CoG and the surface of
the loading, would be equal. Surely there were two ways I could
calculate the position of the CoG, one by the sum of moments and
the other by volume, and the bridge between the two was the fact
that the CoG and the loading surface coincided at the depth I wanted
to know. I sharpened my pencil and threw another log on the fire.
1.
The SUM of the moments on each side of the CoG are equal, ie if
you break the dice down into sections, for each of which you can
calculate the position of CoG and the mass, then
2.
If you consider a hollow loaded dice as being composed of just 2
parts, the shell and the loading material, then when the loading
is negligibly heavy, the CoG of the dice approaches that of the
shell, ie the centre of the dice. When the shell is negligibly heavy,
the dice's CoG approaches the CoG of the loading material. In other
words the dice's CoG will move between the CoG of the shell and
the CoG of the loading depending on the relative masses of the shell
and the loading. In fact in inverse proportion.
Both
equations, 1 and 2, can be expressed in terms of "s",
"w" and "r" (size of side, thickness of wall,
relative density), therefore I had 2 ways of expressing the position
of the CoG, call it "d" for depth, and at optimum loading
point, they were equal. Surely some simple maths would resolve out
into a formula with d on one side and the other variables on the
other. I dusted down my school algebra, sharpened my pencil, and
threw another log on the fire.
As the days passed, it would be hard to say if I consumed more wood
on the fire or in pencils. Quadratic equations came and went, hopes
were raised then dashed, the equation would not be resolved in terms
of d. I quickly found several practical methods of calculating it
by "homing in" on the optimum depth. Here's one:
First
calculate the actual relative density (= weight of dice material
divided by weight of a similar volume of loading material), compare
it to the theoretical relative density given by an estimated depth
according to the formula
then
adjust “d” and repeat until they match to 2 decimal places.
I could
have stopped then, I had a practical and accurate solution to my
problem, lots of other work needed my attention, my supply of pencils
wasn't infinite, I wasn't even sure if the 2 "different"
methods I was using to calculate the CoG weren't really 2 sides
of the same coin, yet I couldn't leave it alone. I was convinced
that there was a way of expressing it which would give the answer
with one calculation. I soldiered on. Fran, my long suffering partner,
became an algebra widow. I would wake in the night, jump from the
bed with "Eureka" on my lips, only to be cruelly crushed
yet again by remorseless mathematical logic.
Then, after 3 weeks, just as I was despairing of ever seeing the
sun again, I had a breakthrough. A brainwave. Eureka (again). I
reviewed my notebooks, organised my thoughts, summarised my workings,
sharpened my pencil one last time, put everything in writing, then
posted it to my nephew Vincent. The one with the degree in mathematics.
I received a reply by return of post. What I'd failed to do in 3
obsessive weeks he'd achieved whilst watching something called Holby
City. I'm not familiar with that particular TV show but I presume
it must be a saga of epic length. He included his workings, which
were all very clear up to the phrase "gradient of the function,"
and then my head began to hurt. He was tactfully vague about my
own efforts. Suffice it to say that he'd done the job, I got my
life back, and henceforth the universal dice loading calculation
shall be known as "Vincent's Formula." I reproduce it
below as my contribution to the sum total knowledge of humankind,
and proof that there's more to being a loaded dice manufacturer
than just immense riches and fame.
Vincent's
Formula
(the universal dice loading formula)
(Please
note that the symbol "d" has a different meaning in this
formula than in earlier ones)
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