Taming the bell | … making probability bell curves more ‘swing-ish’

I only realized something about multi-dice probability bell curves recently … (if you’re a maths-wiz, click away now as the following will probably only offend your sensibilities!).

Bad intuition
Instinctively, I believed that for example 2D6 has the same probability profile as D8+D4. That is, they both give numbers from 2 to 12 and each number on each die is just as likely to turn up.

But, I was quite wrong.
:O|

It turns out the bigger the separation in the die size, the “flatter” the probability bell curve becomes. That is, the more unpredictable the probability spread becomes. In retrospect, this makes sense.

What am I talking about and why could this be useful?
Below is an example where the number range 2 to 12 is generated using two dice, i.e. 2D6, D5+D7, D4+D8, D2+D10 and D1+D11:

In the Range 2 to 12 using two dice (made via anydice.com); the penultimate graph should have been labelled D2+D10 of course!

As above, but presented graphically

So, it can be seen from the above that as the difference in the die size increases, the flatter the probability bell curve becomes (this means the probability curves is becoming more “swingish”).

This is another way to represent the ‘flattening’ of the bell curve; a D4+D8 has a much bigger flat top section (range 5-9) section than 2D6 (just for 7).

Broader plateau seen in top D4+D8 vs bottom 2D6 covering the range 2-12 with two dice

Of course 2D6 systems are widespread in RPGs, for example see a recent post by Larry Hamilton concerning this: https://followmeanddie.com/2019/09/21/newfound-appreciation-of-the-2d6-table/

Here’s another example but for the range 3-18 using three dice (e.g. as used for PC stats, or the damage of a two-handed sword in AD&D).

In the Range 3 to 18 using three dice (made via anydice.com)

As above, but presented graphically

So the message here is, when summing two (or more) dice, you can make the outcome more “random” (less constrained by the bell-shaped curve) by using a big spread in dice size. Edit: it has been pointed out to me that the term “random” as used here is better described mathematically as “variance”.

I think you mentioned useful?
In the main, this observation is not going to change your game. But, this phenomena is worth remembering for the old DM tool kit.  Below are some theoretical examples where this could come into play:

• Swap 2D6 random encounter tables for D8+D4 random encounter tables – instead of one monster being the most common (i.e. on a roll of 7), it’s a set of 5 most common monsters (i.e. rolls of 5-9 are equally common).

• Perhaps the fighter is drunk (and she is fighting with a two-handed sword; normally doing 3D6 damage), being drunk she is less in control of the weapon, and so is more unpredictable with it. Or, perhaps she is on her last hit-point and wants to try a ‘do-or-die’ attack. In either of the above cases, maybe the damage she does could be more unpredictable, more likely to be a flop or a whopper. So, to simulate either of the above, perhaps replace the normal (more predicable) 3D6 (3-18) with the more ‘swing-ish’ D10+2D4 (3-18) …

• 3D6 is also synonymous with rolling up PC stats e.g. STR, DEX, CON etc. But, if in your game you wanted more ‘swingy’ stats, then perhaps again consider using D10+2D4. Again, maybe let the player decide if they want to roll one or more ‘swingy stats’?

• 3D6 or 4D6 attribute tests – some gamers make ability checks using nD6 instead of a D20 (incidentally I compare the two systems here); if you wanted to make the nD6 test less predicable, use pairs of D4+D8 to replace pairs of 2D6s etc.

• Falling damage. Maybe falling down a large tree should be more unpredictable that falling off a cliff. Perhaps the branches might cushion the PC’s fall, or give the PC an extra thrashing on the way down?

• Perhaps you think fireball magic should be more unpredictable than is governed by standard bell curve generated by nD6. Again, mix it up.

• Recently, I’ve found a D6+D8 gives a better probability spread (for a Hex Flower I’m working on) than 2D7 would (which is lucky as I don’t own any D7s).

etc …

Take-home message
This observation is unlikely to change your game much, but ….

… if you get into the situation where a bunch of dice are being rolled and added, and you decide that you want that outcome to be less constrained by the bell, i.e. more “random”, then see if you can substitute the homogeneous dice with dice with bigger gaps between the dice size.

– – –

Me on DriveThru; at the moment I’m mainly pimping my procedural:
:: High Seas ‘Hex Crawl’ – In the Heart of the Sea,
:: Wilderness Hex Crawl – In the Heart of the Unknown,
:: Dungeon/network generator – In the Heart of the Delve & Dangerous

Tuesday Toot!! | ‘God Call’ an Excel Widget

Tuesday Toot – G+ is closing closed. When it was alive things happened. Things unexpected. Great things. Whilst my creative output is only modest, I thought I’d hold something up into the living light, something that came about purely because G+ existed … This is a toot to G+. (published a day early this time …)

This Widget was put up on G+ and automates the process I submitted to ‘& Magazine’.

Background:
The idea is that maybe a PC wants to make a plea for divine intervention … when desperate. This widget is meant to enable that process.

But, in this system, for there to be even a chance for a deity to hear you, the PCs are really going to have to make a sacrifice, a level sacrifice (so it’s going to hurt). Also, there is a chance that if the PC goes ‘chips all in’, the deity could take the PC soul as payment. What can I say, trifle with the Gods at your peril.

And, even if your deity hears you, are they going to act, and act in your favour?

Well, in this system that depends on the nature of the deity (a chaotic good deity might be a helpful sort, but what of Chaotic Neutral or even true Neutral deity?) and what you want from them.  Is your request inline with their ‘legal ethos’ (i.e. lawful, neutral, chaotic) and their spiritual ethos (i.e. good, neutral, evil). If inline with those ethoses, the chance of success is greater.

The Excel widget does the maths … and gives a verdict.

Screen capture:

& Magazine
This widget is based on ‘& magazine’ issued 14; pages 35-40;  “Making the ‘God Call’”. Link: http://www.and-mag.com/wp-content/uploads/2018/02/and-mag-i14-p2.pdf

Widget
The Excel widget can be found here:  Download

– – –

Me on DriveThru; at the moment I’m mainly pimping my procedural:
:: High Seas ‘Hex Crawl’ – In the Heart of the Sea,
:: Wilderness Hex Crawl – In the Heart of the Unknown,
:: Dungeon/network generator – In the Heart of the Delve & Dangerous

Session Zero – ‘Consequences-Style’ Background Builder

Download a (readable)  .pdf version

Idea 
I had this idea about making a Consequences-style ‘session-zero’ background builder.

Visual version of the ‘Consequences’ game

You must have played the “Consequences” game at some point. A group of people take turns to answer a series of questions on a sheet of paper, without seeing the previous answers, and so a random narrative is generated as a result (for more see here).

I’ve come up with my RPG version of this (link above). The idea is that each PC gets a randomly generated background by using the ‘Consequences’ type mechanic at the table. I suppose it’s not necessary for everyone to have their PC’s background generated this way (but that would be more fun), as long as everyone doesn’t mind chipping in to help fill out the questions.

In my version, people get to read the answer (only) directly above the box they are filling in, just to aid with the consistency of the narrative.

The idea is that the answers to the questions can be used to ‘seed’ a background, or used as is (maybe needing a bit of a narrative to weave things together – i.e. to join up the dots).

Here’s a screen-capture of my Consequences back-ground builder:

.pdf version

There are lots of creative people out there who could run with this idea, and I’m sure could make up some (more?) interesting questions, and/or questions that fit their campaign better. Indeed, I’d be surprised if this idea has not been proposed by someone before.  

– – –

Me on DriveThru; at the moment I’m mainly pimping my procedural:
:: High Seas ‘Hex Crawl’ – In the Heart of the Sea,
:: Wilderness Hex Crawl – In the Heart of the Unknown,
:: Dungeon/network generator – In the Heart of the Delve & Dangerous

Tuesday Toot!! | Globe of Futures Forestalled

Tuesday Toot – G+ is closing gone. When it was alive things happened. Things unexpected. Great things. Whilst my creative output is only modest, I thought I’d hold something up into the living light, something that came about purely because G+ existed … This is a toot to G+.

The Master’s Hoard is a compilation of quirk-some magic items which I posted on my G+ page over a period of weeks. The compilation of magic items can be downloaded for free here: Link

:: Globe of Futures Forestalled ::

Summary
An item that gives a player a chance to undo an event

Detail
The globe shows the viewer various fates they may wish to avoid.

Armed with this knowledge, the character can make different life choices, so as to avoid an event (for an adventurer, the event to be avoided is usually their death).

Essentially, the item gives a player a “mulligan”.

The way it works: if the player dislikes a gaming event, the player needs to tell the GM that this was the event their character witnessed in the globe, and the character wishes to avoid that event.

To avoid the event, the player must tell the GM how far back in time the character chose a different path, (thereby the character’s timeline diverging from the fate seen in the globe), this can be up to 24 hrs of in-game time.

For example, the GM announces that a boulder falls on the player character killing her. The player then tell the GM that this was the event the character foresaw in the globe and that she chose a path to avoid this fate. The player elects to go back in time only a few minutes. This time, when she heads down the passage she knows about the trap (i.e. as seen in the globe) and steps around the pressure plate.

Please note this is not time travel per se, the events being ‘rewound’ never happen, they are merely the events as seen in the globe (and now being actively avoided).

The character could go back in time as little as one combat round, e.g. to re-roll an attack they fumbled, or they could go back to a point in time before the start of the dungeon (as long as 24hrs has not expired).

Obviously, all gold, XP, items etc. gained during the period of time being ‘rewound’ are lost. However, things lost (e.g. HPs) are regained. This applies to the whole adventuring group.

Clearly, ‘rewinding’ time poses some logistical difficulties for the players/GM. The players and GM should do their best to get the characters (and the dungeon) back into the state they were prior to the moment time divergence. The player and GM can try to agree a convenient divergence-time that makes most sense. Any discrepancies should be attributed to the vagaries of quantum-time-shivering :O/

It’s important to note that the information obtained from the globe can only be used once. This is because, from the moment that the timelines diverges, the information from the globe is defunct. The globe’s information now concerns the timeline that is now being avoided. Therefore, even if everyone in the adventuring party looked into the globe, only one person can make use of the information.

Looking into the globe more than once leads to insanity without any benefit.

The Globe of Futures Forestalled is a unique item, often kept in the highest of temples and tended to by great seers. The last time it was seen, it was in the hands of the Medusa Oracle.

– – –

Me on DriveThru; at the moment I’m mainly pimping my procedural:
:: High Seas ‘Hex Crawl’ – In the Heart of the Sea,
:: Wilderness Hex Crawl – In the Heart of the Unknown,
:: Dungeon/network generator – In the Heart of the Delve & Dangerous

PC Death is Power Creep … failing forwards in D&D

Edit: I’ve move this sentence to the top: –> If you bust the PC down to 1st level with only basic equipment (and I’m not against that), then there is no need to read on …

I’ve often wondered if PC death leads to party advancement …

Borrgat’s eulogy was brief  …

:: Often when a PC dies this happens

1. PC dies

2. Party (maybe) sheds a tear

3. Party loot the body of gold, armour and magic items, and leave the body for carrion crawlers

4. Dead PC’s player gets new PC; usually at around-ish party level and equipped with new magic items

5. Net result, the party’s position has improved

:: Suggested ritual disposal mechanic

1. (as above)

2. (as above)

3. Party need to ritually dispose of the body

4. All gold and/or XP spent ritually disposing of the dead PC goes into the new PCs XP/GP pot spend. For example, a lavish burial inclusive of burying the PC with all their in life possessions and laid to rest on a bed of  gold and +1 swords

5. The total GP/XP pot for the new PC should probably not surpass the dead PC’s XP total at death (‘power gamers’ would be on this loophole like a whippet)

6. Net result party position is not improved by the PC’s death

Pros:

• Dead PCs are treated more than as a meat-popsicle loaded with loot
• Reduces power creep based on PC death
• Good way to consume excess gold (PCs could even save for a lavish send off)
• Good way to remove all those excess but redundant magic items the players hoard, e.g. the proverbial +1 sword of just-in-case

Cons:

• Players might think you (the DM) are being an arse
• Dead PC’s player might be resentful if the other players don’t give up the loot to bring the PC up to spec …
• Not so easy to generate a ‘decent’ replacement PC in the middle of the dungeon (but then again, maybe the PCs should be thinking of properly disposing of their dead comrade, not pressing on for more dungeon loot … )

Example 1:

7th Level Fighter JUBELO (taken from AD&D’s Tomb of the Lizard King)
dies and is is buried will all his possessions:

• shield +2 (500 XP;  5,000 GP)
• long sword +1, Nine Lives Stealer (1,600 XP; 8,000 GP)
• potion of super heroism (450 XP; 750 GP)
• horn of Valhalla (bronze) (2,000 XP; 30,000 GP)

Totals = 4550 XP + 43750 GP = 48300 XP/GP

So from the above, a DM has three choices to give GP/XP to the player to spend  on the new PC (depending on how ‘hard’ the DM feels about PC death):

i.e.: 4.5K, 44K or 48K XP/GP pot.

For reference, the 1e PHB XP table is reproduced below for fighters:

Example 2:

7th Level Cleric AZURE (also taken from AD&D’s Tomb of the Lizard King)
dies and is buried will all her possessions:

• staff of curing (6,000 XP, 25,000 GP)
• plate mail +2 (1,750 XP; 10,500 GP)
• scroll with:
  • divination (800 XP, 2,400 GP)
  • remove curse (800 XP, 2,400 GP)
  • find traps (800 XP, 2,400 GP)

Totals = 10150 XP + 42700 GP = 52850 XP/GP

Again, from the above, a DM has three choices to give XP to the player to spend on the new PC (depending on how ‘hard’ the DM feels about PC death):

i.e.: 10K, 43K or 53K XP/GP pot.

For reference, the 1e PHB XP table is reproduced below for clerics:

Examples – conclusions

First up there is no need to be quite so formal. A DM might simply take account of the ‘ritual burial’ and think “yes the rights have been well observed, I’ll give a new 5th level PC + some modest magic items etc.

But, let’s work through some of the ‘crunch’ mentioned above:

So, in the two above examples (taking the middle XP/GP pot option; i.e . the bolded option), the new PC would have about 50K to spend between (i) XP for levelling and (ii) gold for magic items (assuming the party buried the dead PC with all their possessions).

But, the party might decide to keep back some choice magic items (lowering the total spend to below 50K).

Or, if the DM is only using the XP value of the magic items (i.e. and not also it’s GP value of the item), the party might spend big on the funeral (e.g. 40K GP, assuming they have it), erecting a statue in the PC’s home town (adding to the total pot spend).

Overall –> the way I’d probably do it would be to tell the player they have about 2/3 of the XP/GP pot to use as XP on the new class. Once the class is settled, I as the DM would probably pick out some appropriate class-related magic items (i.e. using the remainder of the XP/GP pot). Of course, there’s no fun as a player in picking your own magic items.

Like less ‘crunch’ –> give the player a new PC which is about 3/4 the level of the rest of the party (or one or two levels lower). Give the PC no magic items, or 1 item per 2 (or 3) levels of new PC experience.  My guess is that many DM’s instinctively use this method, but without requiring a ritual burial and/or the loss of any ‘party owned magic items’.

However, by contrast, if the DM had simply gifted the party a replacement PC at 7th level with equivalent magic items, the party would have gained probably about  5-10,000 XP and 20-40,000 GP worth of magic items alone. This, just for dying. In some cases, arguably, this would be a better haul of loot than the loot in many a dungeon …

Nonetheless, no matter what I say above, may be the ‘economy of PC death’ in your game works fine, and death is not a shortcut to net party improvement … or perhaps a natural check to a too hard scenario … in which case, of course, ignore all of the above.

– – –

Me on DriveThru; at the moment I’m mainly pimping my procedural:
:: High Seas ‘Hex Crawl’ – In the Heart of the Sea,
:: Wilderness Hex Crawl – In the Heart of the Unknown,
:: Dungeon/network generator – In the Heart of the Delve & Dangerous

Tuesday Toot!! | … drawn in 10 lines or fewer

Tuesday Toot – G+ is closing closed. When it was alive things happened. Things unexpected. Great things. Whilst my creative output is only modest, I thought I’d hold something up into the living light, something that came about purely because G+ existed … This is a toot to G+

Background: For a while, I’ve been interested in the idea of drawing things using 10 lines (or fewer), while still hopefully capturing the essence of the thing …

These drawings will not win any awards, but nonetheless here are some I posted on G+ back in the day:

:: Umberhulk ::

Look into my eyes, wherever they are …

I think I might do a few more of these 10 line drawings going forward …

– – –

Me on DriveThru; at the moment I’m mainly pimping my procedural:
:: High Seas ‘Hex Crawl’ – In the Heart of the Sea,
:: Wilderness Hex Crawl – In the Heart of the Unknown,
:: Dungeon/network generator – In the Heart of the Delve & Dangerous

Combat Morale Tracker | … a simple Hex Flower morale tracker/AI

Edit: A higher quality .pdf version can be found here: Link
and a template Hex Flower Engine can be found here: Template

Combat is more interesting and faster when the morale of Monsters and Henchmen are taken into consideration. This (hopefully) light-weight morale tracker using a Hex Flower Engine should not bog down combat  but also allow moral to be tracked easily.

Background

If you’ve got no idea what this post is about, the below links give some context:

• Link to the ‘theory’ (or at least rational) behind this idea: 2D6 + 19-Hex Power Flower
• Link to more examples of 19 Hex Power Flowers (with some being quite silly/fun): Hex Power Flower (HPF)
• Link to the original posting about using the 19 Hex Power Flower as a Hex Crawl Engine: HPF – What the Hex’s Next?

– – –

Me on DriveThru; at the moment I’m mainly pimping my procedural:
:: High Seas ‘Hex Crawl’ – In the Heart of the Sea,
:: Wilderness Hex Crawl – In the Heart of the Unknown,
:: Dungeon/network generator – In the Heart of the Delve & Dangerous

Tuesday Toot!! | Helm of Sound Sight

Tuesday Toot – G+ is closing gone. When it was alive things happened. Things unexpected. Great things. Whilst my creative output is only modest, I thought I’d hold something up into the living light, something that came about purely because G+ existed … This is a toot to G+.

The Master’s Hoard is a compilation of quirk-some magic items which I posted on my G+ page over a period of weeks. The compilation of magic items can be downloaded for free here: Link

:: Helm of Sound Sight ::

Summary
This item allows the wearer to ‘see’ sound

Detail
The helm functions in 4 main ways:

(1) Passive mode – the wearer can literally see sound waves, and where they are coming from. The sound waves are superimposed on the wearer’s normal vision.

This mode is only really effective in relatively quiet environments. When it’s too noisy the sound waves become indistinct. Seeing sound waves is not the same as hearing them.

The wearer needs to actively concentrate on the sound waves to see them properly.

(2) Active mode – the wearer can use echo‑location (sonar) to see around them. To use this mode a faceplate is flipped down/up/across obscuring the wearer’s face, such that they cannot see properly in a normal sense.

From a crest on top of the helm, high pitched clicking sounds are emitted. These sounds cannot be heard by most creatures. Like a dolphin (or bat), the helm uses the reflected sound waves to generate an image of the surroundings. In this case, the helm generates a monochrome image on the inside of the faceplate/visor.

In this mode, the wearer can see clearly for about 30 feet in a 180 degree arc. Vague shaped can only be made out in the range 30 to 60 feet. Nothing can really be seen beyond 60 feet.

Because the helm uses sound to see, it can be used to see in the dark (even magical darkness), and will pick up invisible or camouflaged creatures.

In this mode, the wearer is immune to ‘stare’ attacks, e.g. petrification gaze.

(3) Probe mode – once a day, the wearer can generate a short-range focused beam of penetrating sound. This allows the wearer to see into things, a bit like x-ray vision, i.e. about 20 cm into flesh, 5 cm into wood and 1 cm into metal or stone.

(4) Blast mode – alternatively, the strong pulse of sound used in (3) can be used to nudge things, range 3 feet, with a force equivalent to that of a finger poke.

– – –

Me on DriveThru; at the moment I’m mainly pimping my procedural:
:: High Seas ‘Hex Crawl’ – In the Heart of the Sea,
:: Wilderness Hex Crawl – In the Heart of the Unknown,
:: Dungeon/network generator – In the Heart of the Delve & Dangerous