Non-Transitive Dice
singingbanana.com
singingbanana.com
We don't have to limit our choices to those two options, and there is a better third one, which undermines the argument before it even got started.
But even if we did limit ourselves to these two options (and I cannot stress enough that there is absolutely no reason to do so), one can still argue that "completely directionless" is a better approximation: evolution has no agency or goal to work towards, it just selects for whatever happens work in any given moment/context. It is the stability of the context that is providing the selection criteria that causes a consistent "direction" to emerge, but as soon as you introduce this explanation we've basically already picked our third option that I mentioned.
I agree, I feel like James is the one that gave Numberphile its personality to me, more so than other mainstays like Matt Parker (I feel like Matt's stamps is more clearly seen in his other projects).
He adds an intrigue and an enthusiasm to Maths that reminds me of how XKCD got me into the subject circa 2006. If anyone is looking for the secret of how to make Maths 'interesting', they should definitely use James and Numberphile as a case study.
That, coupled with the fact that it doesn't matter how MUCH you win or lose by. A loss by 1 and a loss by 4 counts for the same. This nonlinearity is what allows the trick to work, supposedly defying the probabilities.
The “trick” here is only that people assume that dice A winning over dice B and dice B winning over dice C should mean that dice A should also beat C, which it doesn’t simply because dice outcome comparisons aren’t transitive.
So there’s no tricks or secrets, it’s just peoples intuition being wrong.
I personally prefer the intransitive set of D3s to explain this. Assume you have three dice R,B,G. They all have 1,2,3. But on ties we have tie breaker R>B>G on 1s, B>G>R on 2s and G>R>B on 3s. So it’s easy to see that for each dice there is 1/3 of times when it wins the ties, 1/3 where if losses, and the last 1/3 it either wins or loses based on which dice it’s facing. That might at face value seem like cheating because “I put the non transitive property into the tie breakers” but that’s just to make it easier to grok, you can change the dice to have faces (1,6,8),(2,4,9),(3,5,8) and you get the same without needing to handle ties. And since 1,2,3 are all equivalent in comparison to anything above 3, it’s practically the same as the first scenario.
I don't think it works, (1,6,8) seems balanced with (3,5,8).
These are the rows of a magic square.
I picked up a set of these dice from the official site a while back and they were fun to play with.
As board gamers, we also picked up some of the dice that you can roll to determine who goes first. (Dice have various numbers, each die is unique so no ties, but equal chance to have a higher number than the other.)
In general there are 6^n different outcomes and n! different orderings, so a necessary condition is that 6^n is a multiple of n! That means any n ≥ 5 won’t work.
I don't know whether any of those can be non-transitive, though. I'd guess that it's sometimes possible but it's only a guess.
The dice they have for sale: https://mathsgear.co.uk/collections/dice/products/go-first-d...
(No affiliation, just the same site from the original article.)
The first was the non-transitive Grime dice, the second was the board game/Go First dice.
My last comment was only in regards to the second set of dice, that I thought they asked about.
For the Grimes dice, unless I've made an error, it would seem not. I get `(Sum {getSum = 1416},Sum {getSum = 2160},Sum {getSum = 2160},Sum {getSum = 1440},Sum {getSum = 600})` from the following program:
{-# LANGUAGE LambdaCase #-}
import Data.Monoid
oneIf x = Sum $ if x then 1 else 0
toRed = \case
1 -> 4
2 -> 4
3 -> 4
4 -> 4
5 -> 4
6 -> 9
toYellow = \case
1 -> 3
2 -> 3
3 -> 3
4 -> 3
5 -> 8
6 -> 8
toBlue = \case
1 -> 2
2 -> 2
3 -> 2
4 -> 7
5 -> 7
6 -> 7
toMagenta = \case
1 -> 1
2 -> 1
3 -> 6
4 -> 6
5 -> 6
6 -> 6
toOlive = \case
1 -> 0
2 -> 5
3 -> 5
4 -> 5
5 -> 5
6 -> 5
main = print $ mconcat $ do
red <- toRed <$> [1..6]
yellow <- toYellow <$> [1..6]
blue <- toBlue <$> [1..6]
magenta <- toMagenta <$> [1..6]
olive <- toOlive <$> [1..6]
let first = maximum [red, yellow, blue, magenta, olive]
[ ( oneIf $ first == red
, oneIf $ first == yellow
, oneIf $ first == blue
, oneIf $ first == magenta
, oneIf $ first == olive
) ]> This set of five dice is similar to other sets of dice we have seen, in that we have a chain where Blue>Magenta>Olive>Red>Yellow>Blue.
> However, we also have a second chain, where Red>Blue>Olive>Yellow>Magenta>Red.
> Notice the first chain is ordered alphabetically, while the second chain is ordered by word-length.
However on that page I see indeed green, not olive.
Another poster explained the likely reason for this choice of word, but the discrepancy between the shown color and its name is slightly confusing.
Immature olives pass usually through a stage when they are yellow-green, but that stage may not last long.
I agree that using as a color term the name of something that may have a variety of colors, is not a good choice.
Nevertheless, the use of "olive" as yellow-green is entrenched in various domains, e.g. the yellowish green mineral that was called "topaz" by the Ancient Greeks (now topaz means something else) and which is called now "peridot" when in gem form, is also called "olivine", due to its color.
I would count down to the next KoL action point I'd get. IIRC, I was a saucerer.
If you are interested in dice and probability I recommend this new game with a very SEO-unfriendly name: Roll https://store.steampowered.com/app/1585910/Roll/
I have no connection whatsoever, just a sad individual who sank in so many hours!
You try to make the borders so for your party that you will lose by a big margin in one district while winning with a small margin in many other districts.
James is great for explaining these kinds of topics, especially with his enthusiasm and way of talking to the camera, something that is true for other Numberphile hosts/presenters. Definitely go check out https://www.youtube.com/numberphile and https://www.numberphile.com/
And in rpg d20, d12 and d10s are pretty common, so I think it could be tweaked for larger effect.
The challenge would be to make attributes matter in a non-binary and fair way (in the obvious d6+attribute vs d6+attribute system attributes only matter when they overcome the threshold).