Shadows Within to make a comeback?

Entirely possible!

When Adam skipped town head for grad school, his GURPS game was abandoned. We completely disbanded our weekly meet-ups for a while. Recently, we all downloaded Dungeons & Dragons Online, a free MMORPG, to play with the computer as DM. However, the system-choking graphics have motivated me to get to work on a second season of Shadows Within, with 40% more preparation! No expected timetable yet, but the announcement will go out when we start. Shortly afterwards, session updates will resume. Expect plenty of closure, and plenty more McGuffins.

A Mathematical Analysis of Exploding Dice

With a normal die roll, the result is either 1, 2, 3, 4, 5, or 6. This will later be generalized to N-sided dice, but for now, the standard six-sided variety will suffice. Some games use a system called “exploding dice.” With this system, rolling the maximum value on a die allows the roller to roll the die again and add the new result to the previous one. There is usually no limit on the number of rerolls.

For example, a player rolls a six. He rerolls the die, which lands on six once again. Once more, he rolls, but this time gets a two. His final result for this roll is effectively fourteen, even though the die has only six faces .

Put simply, expected value is what the average result for an experiment would be if were tried an infinite number of times. For example, if a coin flip was worth one point if heads lands and two points for tails, the expected value would be 1.5 since heads and tails are equally likely.

To compute expected value, the sum of each possibility times its respective chance is taken. For the coin flip example, this would be . That is, a 50% chance of 1 point plus a 50% chance of 2 points.

So the problem to be tackled here today is this: how does the expected value of an exploding die differ from its standard expected value?

The first step will be to find a pattern. Here is a quick comparison of values and their respective probabilities:

x, Pr(x)
1, 1/6
2, 1/6
3, 1/6
4, 1/6
5, 1/6
7, 1/36
8, 1/36
9, 1/36
10, 1/36
11, 1/36
13, 1/216
14, 1/216
…

So how can these be put into an elegant formula? First of all, a simple sum of products will be used:

The factors preceeding each sum are powers of six, so they can be consolidated into a nested sum:

Each sum, such as , is equal to the average value times the number of elements. The number of elements remains a constant 5, but the average value is increased by 6 for each subsequent sum, beginning with a value of 3. The formula can now be rewritten like so:

Simplifying some of the products, the formula can be rewritten once again:

Now an obvious pattern begins to develop. By expanding these nested products, the pattern becomes even more clear:

Six is factored out of all terms:

This new formula can be rewritten in sumation notation:

For each term added, our error will be reduced by a factor of 6, so just a few terms will return a very accurate result:

N, sum to N
0, 2.5
1, 3.75
2, 4.0972222
3, 4.178240741
4, 4.194058642
5, 4.197595165

The sum is swiftly approaching exactly 4.2, but what is the relation of that value to a six-sided die’s normally expected value?

Adding the exploding modifier to a six-sided die changes its expected value by a factor of . That is, the ratio of the number of sides on the die to the number of sides that do not “explode.”

I have performed the same calculations on four and eight-sided dice to compare results, and the forumla holds.

For any N-sided die numbered 1 to N with all sides equally likely, the exploding modifier will increase the die’s expected value by a factor of .

Interestingly enough, this formala holds for a hypothetical one-sided die. Heuristically, since the highest number will always be rolled, the total will constantly increase, returning an infinite result. Mathematically, the expected value of 1 will increase by a factor of , which approaches positive infinity from positive values.

Silk Road – More Details

Alright, now for a little more in-depth look at the new campaign.

The Characters

Ecthelren: played by Will. This isn’t too much of a shift from Will’s rogue Kerwyn in our D&D campaign. He’s monetarily minded, keeping a close eye on the party’s funds. His main responsibility is to add to the illusion that we really are just a bunch of travelling merchants. Of course, he has no reservations about making some money on the side along the way.

Edrick Dorban: played by Eric (me). Since I was in charge of the last campaign, this is my first time on the other side of the GM screen. Edrick is a practically minded scholar with a penchant for gadgets. He’s an avid hunter, so he’s not completely useless in combat. Adam decided my character can spend time during the trip upgrading his crossbow, making him more of an asset if things get rough.

Chewba Ka: played by Jon. This is very different from Jon’s paladin Gante. Though his new character is still a more combat-oriented one, Jon is playing as a Grond, a hairy creature that can’t speak the common language besides a few broken phrases. Essentially, he’s Chewbaaca. Hence the character name.

Cato Peppi: played by John (this is going to get confusing). John was brought in to our gaming circle by Adam. His character is probably what keeps most of the intra-party communication going, as he’s easily the most fleshed out character; John is probably the only one of us with any decent role-playing abilities. So, his slow-witted farmer character has quite a bit of personality, in addition to unusual quirks.

The World

Adam has done a great job of creating his campaign world. He already knew plenty about Roman society, so he has the benefit of being able to pretty much duplicate that. What’s more, he’s set up a nice map for us using the Google Maps API. Check it out here, or the regular image below.

Silk Road – Introduction

I’m doing what I can to keep updating sans computer, even if it means copying and pasting directly from Adam’s campaign opening lines.

The Domiran Empire occupies a large swath of land around the Straits of Armara in the middle of the Centralian Sea. Several other smaller tribes and city states also cluster around the coast of the sea and its islands. Rumors of magic and strange beings abound, but skeptical minds find scarce evidence for either in the lands near the Sea. The use of iron, writing, and animal power are widespread among more civilized groups.

The current Emperor, Aral V, is a good ruler: strong, honest, and intelligent. He has contacted various resourceful people and has now gathered them in his throne room.

“You have been gathered here today because the Empire has a problem, and you may be able to help me solve it. You all know of the Silk road, which leads from the western provinces off deep into the continent. We trade for silk, spices, and other exotic goods along this road, and as a whole send nearly 50 million sesterces a year back down the road in return.

“And yet I know nothing of the lands where this money is going, and nothing about the origin of these spices and silk. The writings of historians give reliable, reasonable sounding reports out to the port city of Tarpa in the land of the Gratians, but after that they trail off into traveler’s tales of headless men, singing trees, and giant birds.

“I want information. Reliable, first hand information on the entire road from here until wherever it goes. I want to know where the Empire’s money is going, what the secret of silk is, and anything else useful, or even just interesting. Accomplish this task and you will all receive titles and fertile land in one of the new provinces.

“Since you are traveling the silk road, I suggest you disguise yourself as merchants. It might be unwise to let the eastern lands learn that the Empire is looking their way. Provisions from the Imperial storehouses will be made available to you, and a cipher book for encoding reports. Send them back down the road with instructions to hand them to the garrison at Pemra. The master of provisions in the antechamber will take you to your supplies.”

No updates for a bit

The laptop is once again out of commision. Posts will resume next week, after which I should have my new computer.

A whole new world

As I mentioned earlier, Adam’s campaign is up next. We’ve already been playing it for a few weeks, learning this strange and foriegn GURPS. We also have one new player who’s been adding some much needed dynamics to an otherwise boring team.

One of the things that annoyed me about my campaign was that the players went through the game without much inter-character interaction. I mean, I’m glad they acted as a team, but there was always a ridiculous level of caution applied to keep the party together (even going so far as to tie themselves together with rope).

This new campaign is proving much more interesting. The characters, who I’ll introduce soon enough, took a more believable amount of time to gain each other’s trust. Their varied backgrounds make for interesting clashes of culture.

The world is a Roman-esque Iron Age setting. The main characters are called together to explore the silk road to the east at the emperor’s request. It’s an intellignce gathering mission, mainly. I’ll go into more detail later, but for now, here’s a quick summary of the PCs:

• Edrick Dorban – scholarly tinkerer
• Ecthelren -  fast-talking merchant
• Chewba Ka -  Grond soldier
• Cato Peppi -  dirt-poor farmer

Shortly after session 69

“Well, that’s it. The dragon’s dead.” The GM flips over the small piece of cardboard in the center of the paper grid.

“About time,” Jon says. “We can make armor out of it or something, right?”

“Oh, definitely.”

“Good. I think my paladin deserves some reward after all that.”

Adam picks up the gnome miniature from one corner of the table. “I wouldn’t mind a cape made from its wings, really. That’d be worth some kind of fire protection, maybe.”

The GM leans back in his chair. “Yeah… I guess we’ll work something out for that.”

“I was still hoping for a treasure hoard,” Will says.

“Heh, is there anything else is a rouge thinks about?”

Notes are jotted as experience points are tallied. “If you don’t mind me asking, what was Murc at, HP-wise?”

“Heh, you guys are gonna love this,” the GM says, looking up from a page of hastily jotted notes. “The poor half-orc’s down to exactly one.”

“Well, I for one am glad he made it through.”

“Yeah, Murc’s pretty surprised he’s alive. He holds his arm up and says ‘we did it? We did it!’” the GM mimes the barbarian’s actions as he describes them.

“So why does Murc have to sound so Texan, anyway?”

“There’s only so many voices I can make…”

“So what about Melrain?”

“Oh yeah… forgot about him…”

***

The players file out from the apartment.

“See you guys tomorrow.”

“Yep.”

“Later.”

The GM and the Roommate are left with empty soda cans and half-eaten bags of chips. The GM nods. “That’s the first campaign I’ve ever run to some sort of actual ending.”

“You went easy on them, didn’t you?”

“What? No! Maybe. A bit.” The GM begins clearing off the table.

“You should’ve killed ‘em off. That’d be a great ending.”

“No, that’s like, an X-Men 3 ending.”

“So what are you going to do with your Sunday nights now?”

“For the next couple weeks… I dunno. Video games? We’ll probably just all hang out until Adam’s ready to start his campaign. He’s going to run a GURPS game set in a Roman tech level.”

“‘GURPS?’”

“Don’t worry about it. I’m sure you’ll learn about it once we get going. In fact, you could join in as a player if you want…”

“Oh, no. You’re not dragging me into your geeky culture.” There’s a short silence in the room, broken by the air conditioner whirring to life. “You really should’ve just killed ‘em off.”

Session 69

Having fled the city, the team now approaches the goblins’ base of operations.

Session 68

Chaos erupts as the reclaiming of the city begins.

Session 67

In a tunnel system beneath the city, the team attempts to loosen the goblins’ grip on Remnas.