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Subject: Dice Roll Statistics rss

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Garry Hoddinott
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I would like to know the percentage outcome of this dice roll mechanic.

3 d6 dice are rolled. What is the smallest possible whole number (ie ignore minus sign. read -4 as 4) difference between any 2 dice summed and the 3rd dice. . Answers will vary from 0 - 6. Six only occurs if all dice are 6 ie (6 + 6) - 6 or 6 - (6 + 6). On a dice roll of 6,5,5 possible answers could be (6+5)-5 = 5 or (5+5)-6 =4 - in this case 4 would be the smallest possible outcome and the players choice as numbers closer to 0 have the most favourable effect. Another example might help Rolled 5,2,2 Answer is 1 as in 5-(2+2)

Its a very hard question I think. Maybe you can point me to where I might be able to find an answer.

Also - perhaps you would like to comment on whether you think this is a useful mechanic. I don't really want the 10 options of roll 2 and add, further - those percentages don't work too well for this game. Rolling one die gives a very limited scale of outcomes.

Can this be expressed as "find the lowest whole number difference between any 3 dice when 2 are summed."
 
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Moritz Eggert
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I would suggest writing down all possible outcomes of the three dice (216 possibilities is manageable):
1,1,1
1,1,2
1,1,3
and so forth
Then write the possible differentials next to the list.
Then you count the number of results for 0,-1,-2 etc. and calculate their percentage in the possible 216 outcomes.
For example if -2 is possible in 50 of 216 rolls the calculation goes
50: 216
equals
x:100 (%)
formula:
50x100:216 = 23% (rounded down)

So you would know that the possibility of -2 is 23% of all rolls.

I am sure one could also create a complicated formula, but for such a small range of possible rolls it is probably much quicker to simply count the instances.

I would do the math for you, but don't have the time right now :-)
 
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Jeremy Lennert
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Here's a program for you:

http://anydice.com/program/e70e

I wouldn't particularly recommend this system, just because it seems like a lot of work to use.

If you want a system with about 6-7 possible output values where rolling high is very rare, you could consider something like "roll 3d6 and take the lowest die" or "roll 6d6 and count the number of dice that roll 5 or higher".

Here's what those look like, for comparison:
http://anydice.com/program/e70f

cool username wrote:
Answers actually vary from 0 to 11 (e.g., 6+6-1)

No, because if you roll 6, 6, and 1, then you're supposed to take 6+1-6 = 1, because you're required to arrange the 3 dice in such a way as to produce the smallest possible result.
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Christopher Dearlove
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Tried pointing out error, but got ninja-ed both by a post and the posting being deleted. The latter makes your post fail, which I did not know.

I have a program which I can do this sort of thing with. And I get


0 - 45 ~ 20.8333% = 5/24
1 - 76 ~ 35.1852% = 19/54
2 - 49 ~ 22.6852% = 49/216
3 - 28 ~ 12.963% = 7/54
4 - 13 ~ 6.01852% = 13/216
5 - 4 ~ 1.85185% = 1/54
6 - 1 ~ 0.462963% = 1/216


This won't make much sense, so I'll hide it.

Spoiler (click to reveal)
r:=repeat3{d6};min{abs(-r0+r1+r2),abs(r0-r1+r2),abs(r0+r1-r2)}
 
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Koen Hendrix
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I would not want to have to do this kind of mental arithmetic after every dice throw. It's both time-consuming, which dilutes that dramatic tension-release of chucking dice, and error-prone.


Anecdotal evidence: I once designed a racing game where players rolled 3d6. If their car was in Low gear, they'd move the lowest rolled value; in Middle gear the middle value and in High gear the highest one.

I playtested it with random people, and it was too complicated for them. They would probably have gotten used to it eventually, but their first play experience was pretty bad.

I binned the movement system after that. Much better to just have different dice for different gears (even if it costs a bit more component-wise). Ease of play trumps mathematical cleverness.



Jeremy's alternative suggestions are excellent.

You could also roll 4 dice and add up the lowest two. That'll give you a range from 2 to 12, with the low numbers being far more likely.
http://anydice.com/program/e715
 
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Christopher Dearlove
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khendrix wrote:
I would not want to have to do this kind of mental arithmetic after every dice throw.


Don't try playing Can't Stop then.

That said, I don't think it's a great system. But in large part the difficulty depends on the players.

If I roll two dice, and let's say I get a six and a three, I don't see a 6 and a 3. I see 9. (Of course I also see 6 and 3, but that's secondary if a total is what I want.) That is with standard spotted dice. With dice with digits on them it's slower, and maybe I add the digits (quickly enough only I would notice).

Three dice? Not as practiced. But if I'm back in the saddle, so to speak, in the middle of a game of Kingsburg, I can do it. (And if I make an error, it's as likely or more to be by 2 as 1, because I'm pattern recognising, not adding.)

My point? It's familiarity. While I avoid Monopoly like the plague, there are enough 2d6 games out there to be familiar. I strongly suspect if you tried it on RPG players (especially of ones with their own funny systems) they'd do fine. Mass market? Forget it.

Incidentally I like the three gears, to me it's trivial. But I'd be wondering about the distribution (as I do playing Automobile).
 
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Garry Hoddinott
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Christopherr=repeat3{d6};min{abs(-r0+r1+r2),abs(r0-r1+r2),abs(r0+r1-r2)}
Perfect example of the beauty of Math. I kinda get it but would dearly love to be able to think with the clarity it takes to have a full grasp of this. Thanks.

"Don't try playing Can't Stop then". I agree with your sentiment here, its all about practise to quickly recognise solutions. I tried the 3d6 thing with my very non gamer, non mathy wife. Within 10 throws she pretty much arrived at the lowest solution quickly. I had played with it a bit and thought through it a lot and found the solutions popped out at me very quickly. Can't Stop mechanism is really neat


Koen: I'm amazed people thought the gear selection solution for your racing game was too hard, it seems extremely logical and a very good thematic fit.


Moritz: Sorry 216 combos is about 200 more than I would try to do manually - I'm a lazy bum, and it kind of defeats the beauty of the math answer. Well that's what I reckon anyway.

Ok .. wanna know why I was interested in this. Well .... you need to go back to Ancient Greece, the Legend of Atlanta and the Golden Apples. Its a fun story and has a moral as most of them did. Anyway, she is a very fast runner and discourages suiters by challenging them to a race. If they win she marries them, if they lose, they die! Hippomenes falls in love at first sight but ain't up for dying so seeks the help of the goddess Diana who blesses golden apples that will entrance Atlanta and cause her to stop and pick them up. When Hippomenes is trailing he throws an apple to take Atlanta off course. That's what I needed. She has a movement penalty to pick up an apple but I wanted a random element of which side of the track and how far the apples rolls. Which side - Odds and evens - how far - the 3d6 idea I had. Turns out my suspicions were kind of on track.
 
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Alison Mandible
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GarryHoddinott wrote:
Moritz: Sorry 216 combos is about 200 more than I would try to do manually - I'm a lazy bum, and it kind of defeats the beauty of the math answer. Well that's what I reckon anyway.


If you don't relish the idea of doing that procedure over and over again, your players may not want to either.
 
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Christopher Dearlove
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Doh.

Stepping back from the problem and thinking about it rather than jumping in and solving it, the lowest difference of two dice and one die always happens when the one die is the largest.

A significant simplification both doing it and analysing it.

And which can replace previous by:

Spoiler (click to reveal)
r:=sort(repeat3{d6});abs(r0+r1-r2)
 
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