How thick is a three-sided coin?
jasmcole.com
jasmcole.com
D&D turns out to be a reasonably good exercise in delayed gratification, an essential life skill many people need in order to successfully rehabilitate.
Wasn't D&D though, the rules (most of them made up) were particularly suited to that sort of play: lots of storytelling and brutal combat.
You're allowed to do anything that "makes sense" given the fiction, and it is so much more storytelling focused than D&D etc.
You can easily create a reasonable n-gon (for low n) on card or paper and add an axle from wood or whatever is to hand and you have a Dn spinner. D3, 4, 5, 6, 8, and perhaps 10 and 12 will work too - yes more than the usual but why not?
If you have paper and pen and something that will give you heads/tails ie 0/1 then use binary and generate tables of numbers and use them one by one. Other bases and simple devices can be used to get tables for a desired "die" max value.
The whole point of DnD n that is it is fantasy ie thinking and talking about something that is not real. How you deal with fantasy is up to you.
Worrying about formalality is generally not a good approach to dealing with phantasy for obvious reasons.
The dream of the cake vs the actual cake.
It's a really freaky shift of consciousness when you think about it. A whole culture trained to prefer dreamcakes over realcakes. To prefer dreams over reality.
The standard interpretation of the original results was that a skill at delayed gratification led to better outcomes in life.
A replication of the study found that the effect nearly completely disappears once you correctly account for their socio-economic background. [1]
It also strongly supported the criticism that the lower socio-economic kids may have actually been making a logical choice to take the first marshmallow -- many of them had grown up with a lack of reliable adults and a lack of trust that future dreams will come true. Under that worldview, it makes sense to take the bird in the hand over the two in the bush.
Preferring dreams over reality is a luxury mostly permitted to those whose dreams have a reasonable chance of coming true, and a cushion to hand on when they don't.
1. https://www.theatlantic.com/family/archive/2018/06/marshmall...
It’s not a marshmallow test, it’s a wealth test. Surprise, wealthy kids have better prospects.
> So what should the value of h be?
To which the author replied:
> It's about the journey, not the destination
Also, the fit they did for lambda was based on simulations, not actual "coin" tosses. So the exact answer for h/r probably wouldn't be very helpful.
Doesn't seem like it's getting regular updates to the core physics anymore despite a lot of open issues and pull requests (although I do see two very minor git commits from this calendar year).
For 1d3 roll 1d6 and divide by 2.
Edit: apparently for some 1d3 designs it is difficult to roll them fairly: as per https://news.ycombinator.com/item?id=33778669
https://boardgamegeek.com/thread/1673702/roll-1d3/page/2
Is just a six sided die with three edges rounded off.
Edit: Even more generally, you can use any set of fair dice. Imagine you have a d2, d4, and d6, and you want to simulate a d40. Roll the d6 and multiply by 4*2; roll the d4 and multiply by 2; then roll (flip) the d2. The resulting sum is a fair roll in the range (0, 48]. This is a significant improvement over rolling a d6 three times: you can roll all of the dice together, and you're much less likely to need a reroll.
The bit I'm having trouble with...aren't you missing the low end of the range?
Like say you roll a 1 (the lowest possible roll), you'd get, if I understand right, you'd get 14 + 12 + 1*1 = 7.
Maybe this works if you subtract one from each roll so they can roll 0?
So a d4 would have 1 & 2 on the square faces, and 3 - 3 - 4 - 4 on the rectangular faces; this would require flattening the die.
A d5 would have 1 & 1 on the square faces, and 2 - 3 - 4 - 5 on the rectangular faces; this would require stretching the die to even out the probabilities.
I did an analysis similar to this, using the solid angle subtended by the faces, but the "bouncing" model is obviously much more rigorous and I'm curious to apply it to this case as well.
This should extend to any N, do sets of N flips until you get exactly one of each N distinct result, and take the first as the final result.
For example it works for simulating a d3 using a two-sided coin: THH or HTT -> 1, HTH or THT -> 2, HHT or TTH -> 3, HHH or TTT -> reroll.
This can be combined with your protocol, but it takes some thought to interpret because in your protocol the result is what actually shows up on the coin, but in mine it's the position.
1: 0
2: 1/2
3: 7/9
4: 58/64
N: 1 - (N-1)! / N^(N-1)
By the time you have a typical 6-sided dice, you only get 6 different rolls in 6 rolls 1.54% of the time.
I haven't calculated what happens when bias is introduced but my intuition is that with more bias you would discard more. (Based on the extreme case that if one of the sides can never roll you would never accept a result.)
A: No at an microscopic/atomic/quantum level there are trillions of sides
(Note: make sure to give them toes.)
A computer model using a deterministic physics system would be completely aware of the initial conditions of each toss. You'd presumably want to introduce some randomness in those initial conditions (using e.g. a pseudorandom number generator) to simulate human coin tosses.
At least according to Twitter and one of the responses to the first "answered before" source linked in the article: https://twitter.com/radu_nicolau/status/956926727971852290
Coincidentally I recently had a similar issue on a personal project using HTML Canvas that resolved when I discovered I was inadvertently using `ctx.rect(...); ctx.fill(...);` instead of `ctx.fillRect(...)`.
Not familiar enough with Matter.js to know how relevant that might be!
However, the question here is simply directly the question "How thick does a coin have to be to equal chances of landing on its side or either face?", in which case changing the design to something else is simply to answer a different question. At best you might try to extract something useful from the alternate formulation to bring back to your original question, but in this case I wouldn't see any value; we already know that we can construct n-sided symmetric objects from curved faces for n >= 1 (if you consider a sphere a 1 sided die) and it doesn't help the original question much.
I feel the same about some weird d3 shape too, because it doesn't really roll and would require a similar "toss" that's hard to judge. I'd rather have double-marked d6s, which do seem to be available.
The problem with a three sided coin is that it will roll away before you can read it. ;) Increasing friction of the coin so that does not happen is left as an exercise for the reader.
Alternatively, imagine a football shape forced into a triangular cross section.