4 ms·

There isn't actually an equal chance of going first, though, is there?

Someone 5y ago
For 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 ago
  For 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 ago
Why not. If all players/dice have an equal chance of rolling highest, then...?
- dllthomas 5y ago
  If 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 ago
The 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 ago
  Go 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 ago
    Sorry, 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.

For 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 ago
For 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 ago

Why not. If all players/dice have an equal chance of rolling highest, then...?

dllthomas 5y ago
If 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 ago

The 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 ago
Go 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 ago
  Sorry, 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.