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Grimes' appearances on Numberphile are some of my favorites to watch. I picked up a set of these dice from the official site a while back and they were fun to
by james-skemp 5y ago
Grimes' appearances on Numberphile are some of my favorites to watch.
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.)
- deleted 5y ago[deleted]
- dllthomas 5y agoThere isn't actually an equal chance of going first, though, is there?
- Someone 5y agoFor n = 2, a trivial solution is to have one die with three 1s and three 3s and one die with six 2s. 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.
- deleted 5y ago[deleted]
- dllthomas 5y agoFor equal chance of all orderings, yes. "Equal chance of going first" is more flexible, though I expect at some point we hit a limit in expressiveness (at least) if we're still using 6 sided dice. 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.
- verve_rat 5y agoWhy not. If all players/dice have an equal chance of rolling highest, then...?
- dllthomas 5y agoIf that, then yes, of course. But does that hold? 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 ) ]
- james-skemp 5y agoThe math behind it is a little beyond me, but http://www.ericharshbarger.org/dice/go_first_dice.html http://www.ericharshbarger.org/dice/go_first_dice.html The dice they have for sale: https://mathsgear.co.uk/collections/dice/products/go-first-dice https://mathsgear.co.uk/collections/dice/products/go-first-d... (No affiliation, just the same site from the original article.)
- dllthomas 5y agoGo first dice are cool but I don't think they are non-transitive. In particular they claim that each ordering is equally likely, and for any pair of players (Jack and Jill, say) we can observe that across all orderings we should find Jack preceding Jill exactly half the time. This would mean that no pair of dice should exhibit the kind of edge that exhibits non-transitivity in non-transitive dice.
- james-skemp 5y agoSorry, two different lines of thought. 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.
- danw1979 5y agoAlways grateful for a YouTube channel recommendation, but it would have been awesome if it really was the Grimes I thought you were referring to.