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Messages - ycleption

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21
Just for Fun / Sometimes I get bored at Law school...
« on: September 10, 2007, 05:11:35 pm »
Okay, given the overwhelming response and outpouring of accolades generated by the first time I did something like this, and my burning curiosity on the subject, I decided to figure out the exact average of the best three out of four d6.  (For those of you unfamiliar with non-layo D&D, often three out of four regular six sided dice is used to assign a character's abilities)
Sooo... I decided that the best way to go about this was, rather than calculate the roll as a whole, just to concentrate on the discarded die, and figure out the value of the average low die, and then subtract that from 14, which is the average of four dice.
With that decided, the problem becomes counting: I needed to figure out how many times a given value is the lowest die, then multiply that by the value, and add those numbers together:
So, I started at the top, because I knew there would be fewer, and thus it is easier to count.
The first one is real easy... six is the lowest die exactly one time: when all four dice are sixes.
1x6=6
yay.

Now, with five, it gets a little more complicated. To organize things (which will come in handy in a bit), we will separate things by how many fives turn up. (In mathematical terms we are "choosing" successive numbers out of four)
There is one case where there are four fives and five is the lowest number: (5555)
Four where there are three fives (5556), (5565), (5655), (6555)
Six where there are two: (5566),(5656),(5665),(6556),(6565),(6655)
and Four where there are one: (5666),(6566),(6656),(6665)

In math terms:
4 choose 1 = 4
4 choose 2 = 6
4 choose 3 = 4
4 choose 4 = 1

That's fifteen ways (4+6+4+1) that a five is the lowest die, and 15x5=75


Okay, I don't know about the rest of you, but I'm lazy, and counting the fives in my head is about my limit, so I'm going to have to figure out something better to use for the fours and below.

So, if I look at what I've written out in the fives, I can see that the other numbers will follow that pattern. Incidentally, the 4,6,4,1 pattern is part of pascal's triangle (the fourth line, because there are four dice), and crops up in various other things. With 5, there was only one other number to consider, but with four and below, we have to multiply the number of ways the number we're looking at can occur, by all the permutations of however many larger numbers there are. Luckily, permutations is easy to figure out. We just have to take the number of dice we are looking at, to the power of the available slots.

For example, take the specific case where there are three fours: there are two numbers greater than four (five and six, in case you have trouble counting), and one available slot. So, we have (2^1)(4 choose 3) = 2x4=8

We can confirm this by counting them out:
(4445),(4446),(4454),(4464),(4544),(4644),(5444),(6444)
 
Now we have a general rule for four six-sided dice (I have this written in formal summation notation on a post-it note, but don't know how to display math symbols on the forum): The sum from one to six of each value multiplied by the sum from one to four (we'll call it n) of six minus the die value to the four minus n power, multiplied by four choose n.

Because that's a mouthful, (and probably not terribly well stated) here's the actual numbers:

  1[(5^3)4+(5^2)6+(5^1)4+(5^0)1]
+2[(4^3)4+(4^2)6+(4^1)4+(4^0)1]
+3[(3^3)4+(3^2)6+(3^1)4+(3^0)1]
+4[(2^3)4+(2^2)6+(2^1)4+(2^0)1]
+5[(1^3)4+(1^2)6+(1^1)4+(1^0)1]
+6[(0^3)4+(0^2)6+(0^1)4+(0^0)1]

Performing the calculations, 671 + 738 + 525 + 260 + 75 + 6 = 2274
So, dividing 2274 by the total number of possible rolls which is 6^4, or 1296, we get about 1.745
As we said at the top, we subtract that from the average of four dice which is fourteen, so 14-1.745 = 12.245

There's what we've been looking for: the average of the best three out of four d6 is about 12.25

Whew!

Of course, it's no fun to end there.... what if we needed to rollthe best 7 out of 8 d10 instead? We can generalize to the best x out of x+1 dice of any number of faces..
(We could potentially generalize to the best x out of y, but that would need an addition summation, and would be much much more complicated, and it's messy enough as it is. Also, we could make it look a lot neater using summation notation). We'll call the number of faces d, the number of dice n, and designate xCy as the choose function, x choose y.

  1[((d-1)^(n-1))(n)+((d-1)^(n-2))(nC2)+ . . . +((d-1)^2)(nCn-2)+((d-1)^1)(nCn-1)+((d-1)^0)1]
+2[((d-2)^(n-1))(n)+((d-2)^(n-2))(nC2)+ . . . +((d-2)^2)(nCn-2)+((d-2)^1)(nCn-1)+((d-2)^0)1]
+3[((d-3)^(n-1))(n)+((d-3)^(n-2))(nC2)+ . . . +((d-3)^2)(nCn-2)+((d-3)^1)(nCn-1)+((d-3)^0)1]
+
.
.
.
+(d-1)[(1^(n-1))(n)+(1^(n-2))(nC2)+ . . . +(1^2)(nCn-2)+(1^1)(nCn-1)+(1^0)1)]
+d[a bunch of stuff I don't feel like writing out but equals 1]

Wasn't that exciting?
The following users thanked this post: Stephen_Zuckerman

22
Ask A Gamemaster / Finding other people's stuff.
« on: April 12, 2007, 01:06:20 am »
So a few hours ago, I found something on the ground. I asked on the DM channel what I should do, and got no response. I went ahead and picked it up and posted here IC to try and find the owner.
http://www.layonara.com/wild-surge-inn/115291-found-weapon.html

Is there an official line about what one is supposed to do in such situations? I just feel kinda weird about just picking somebody else's stuff up, even if it is with intent to return it.
Thanks
The following users thanked this post: Serissa

23
General Discussion / Partying: a personal request
« on: April 04, 2007, 06:31:38 pm »
I'm sure I'm not the only one who has experienced this:

You invite someone, or someone invites you into a party. Someone else wanders by, initiates conversation, inquires as to your purpose, and joins the party. You are about to set off to smack things. Suddenly, and without warning, another face appears on your sidebar... you mouse over their portrait, and see that they're on another continent . . . Strange. . . . And then, another person is added and another.  More people join (none of whom are actually present) until the entire population of Layonara is in a single party!  All of a sudden, everyone has a reason to wait before setting out, "One minute, it is vital that I stare at my navel before heading into battle." Lo and behold, all those little faces in the party bar just happen to wander by, having the urge to kill the same things at the same time at the same place.Funny how that works.

Seriously though, there is a big difference between:

Me: This is a strong foe we aim to battle. Perhaps I should send a bird to Lady Pwnsalot, to see if she would be interested in joining us.
Other Person: Indeed, her sword would be welcome!
Me: *summons a falcon and writes on some parchment*
[Tell] Falcon message: Myself and a few others aim to slay the fearsome rats in the sewers, but we cannot do it alone, would you care to join us?
[Tell from Lady Pwnsalot]: Verily! I would never turn down such an opportunity, I shall be there forthwith!
*invite Lady Pwnsalot*
Me: *Receives a not from a bird, reads it and nods sagely* She will be here to aid us as fast as her legs can carry her.


And:

[Tell]:Hey, wanna come kill some stuff with us?
[Tell from Lady Pwnsalot]: Sure, I'm in Allindor, just wait there for me, ok?
[Tell] n/p
*invite Lady Pwnsalot*


There are plenty of legitimate reasons why one might invite someone who isn't there at the moment, but please have the grace to ask the rest of the party members, and at least make an attempt to have some RP justification. Too often, this kind of thing just spirals out of control (although my initial description may be a tad exaggerated ;)). I know I'm not the only one who prefers small parties, so please try to remember when everyone wants their friend to come along, it can ruin things for many of us. If you want to have large parties that can run roughshod over anything, there are plenty of people who enjoy that too, but to make every gathering like that is disrespectful to the rest of us, in my opinion. It really doesn't take much imagination to invite people in an RP appropriate way that allows the group to have their input. Maybe character A will refuse to travel if character B  goes along. Maybe Character A is shy and doesn't want to fight with people he doesn't know. And maybe some of us just have OOC preferences, and feel left out when people are invited without some IC dialogue. A little bit of communication and courtesy go a long way.  Thanks.

*I hope this doesn't sound too whiny and ranty, as the title says this is just a personal request, based on my own play style, and I realize not everyone is going to agree with me here*

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