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 die as being composed of just 2 parts, the shell and the loading material, then when the loading is negligibly heavy, the CoG of the die approaches that of the shell, ie the centre of the die. When the shell is negligibly heavy, the die's CoG approaches the CoG of the loading material. In other words the die'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, honed my pencil to a needlepoint, 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, Fran got her man 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)
This formula ignores the corners of the die. The results are highly accurate for thin walls, but less so for thicker ones - a practical solution.
Bored mathematicians please note: I'm still looking for the definitive formula!
Vincent's Formula
