House Rules in a Digital House

I've been running D&D 5e for a group of remote friends recently. These are the “house rules” that I've adopted to make things go smoother, or at least reassure myself. I'm sure I got most of these from other places, but I can only credit what I remember.

Absence

Characters of absent players will be ignored and their departure and return unremarked on. I will make no effort to explain this in the fiction, but in future sessions we can act as though they were there. So “remember that time we all burned down a fortress?” or “aren’t you those people who tried to murder my brother?” will still include characters of players who were absent at the time.

I don't plan to spend a lot of time balancing encounters, so I think “fading into the background” is a fine solution. I want to encourage roleplaying, but also a shared group identity, so this doesn't let anyone off the hook or force anyone to miss opportunities because of real-life concerns.

Experience

Experience will be awarded using the “Milestone” option. While we are in a dungeon, “clearing” a level will be worth one level of experience. Outside of a dungeon, other goals may be used. These are negotiable.

I started the game in a dungeon and expected the first level to be cleared fairly quickly (it was). So this gave us a quick level-up without math on my end, but still tied to a measurable achievement. For the first-time players in the group, the early level-up lets them engage with the rules differently and see possibilities for (mechanical) growth. Because I do not know what parts of the game this group will enjoy most, the “Milestone” option allows them to pivot away from the dungeon later.

Inspiration

Inspiration will be awarded at the end of each session by player vote for favorite moment.

This is something I borrowed from DIE TRYING, although I've seen similar things in other systems. The practical effect is a moment of reflection at the end of the session when players can say what they liked best and what they're looking forward to most. It also reminds me that inspiration exists.

Quorum

We will not play D&D with less than half the party. To keep the appointment, we can play other games with however many people we have.

It feels bad for the players who do show up when there aren't enough people to play, so I really wanted to get in the habit of playing even if we can't play the main game. Games under consideration for backup have to be lightweight. So far they include:

• Lasers & Feelings
• Three Black Crows, Three Dead Men
• Electric Bastionland + Some One-Page Dungeon

Timekeeping

Generally, I will assume an “exploration turn” of 10 minutes, which is enough time to fully explore a room and interact with all of its contents (excluding combat etc.). After 8 hours (48 turns) without rest, characters will take a level of exhaustion.

I didn't find timekeeping rules that I liked in 5e, and as long as we were in a dungeon I wanted to build expectations around that procedure. In practice, the biggest effect of this change is that ritual casting now carries some cost. It does seem like 48 turns is unlikely to be an actual concern.

Encumbrance

I haven't adopted these rules yet, but I'm considering something like Electric Bastionland (paraphrased):

Some items are [bulky]. You can carry two [bulky] items: one on your back and one in your hands.

The hope is that these rules are just enough to make looting things complicated, but not enough that I have to worry about weight or spreadsheets. The worry is that with six characters, this just still doesn't matter.

Hexcrawling

These are also rules still under consideration:

Travel 3 hexes (6 miles each) per day, walking 8 hours per day.
Roll a d6 2x every day of travel, once during the day and once during camp.
On a 1, an Encounter occurs (something interesting).
On a 2, an Omen occurs (signs of something interesting).
Difficult terrain (mountain, swamp) counts as 2 hexes.
Travel -1 hex to explore for hidden locations in one hex.
Travel -1 hex to forage, gaining 1d4-1 rations per forager.
Travel +1 hex on a road.
Travel +1 hex if everyone has fast mounts.
Travel +1 hex over another 4 hours but [take a level of] Exhaustion.

I didn't find any hexcrawl rules in 5e, so I stole these ones wholesale from Moonhop. Because we have new players, I'm trying to fit as many “modes” as I can into the game, to showcase it a bit. At the same time, I'd like to minimize the overhead of learning a hundred new subsystems and then abandoning them later. I might have to make it two levels of exhaustion though, in order for it to matter.

Other Notes

I've observed some other things about the group in particular and online play in general.

• Playing on a weekday evening, and given the limits of engagement online, 3 hours is about as long as I can go.
• Some players fare better than others with “theater of the mind”. There has been a request for a VTT system of some kind, which terrifies me, but I'm considering it. On the other hand, I'd like everyone to be on the same footing, so maybe I can strike a balance with a whiteboard of some kind or some more defined abstractions (“Zones” from Fate or “Abstract Distances” from The Black Hack).
• None of the players has actually used their inspiration yet, so I'll have to be better about reminding them when they can. The post-game ritual seems to go over well though.
• D&D is actually fun! This should be obvious, because why else would be be here? But it's always refreshing to play.

Chuckleheads and The ’Valizzi

Dan D. at Throne of Salt requests from the jackalope:

d6 Chuckleheads you have to deal with from the local wizard mafia

The Plan

This should be easy:

1. Binge-listen to Crimetown.
2. Define the Local Wizard Mafia.
3. Write up some chuckleheads.

Wizard mobsters are probably like normal mobsters, but with funny hats. (Funny hats are by Lorc and Delapouite via game-icons.net, used under a CC-BY 3.0 license.)

The ’Valizzi

Description

The ’Valizzi is the local wizard mafia, an organized magical crime syndicate. Even when their ends are mundane, their means are supernatural. They run extraplanar smuggling rings, deal in apotropaic protection rackets, and arbitrate disputes between wizards and godlings. Where magic is illegal or disreputable, they have business with necromancy, fortune-telling, and the manufacture of consumables (potions, scrolls, drugs, and so on).

There is no requirement of blood relation to join the ’Valizzi, only magical ability. Because of their persecution, proper members of the ’Valizzi are magically bound to never verbally confirm if an individual is or is not a member of the ’Valizzi. This curse is taken voluntarily, and has never been known to be broken. Current members will also never knowingly risk letting an outsider know who is or isn't in the ’Valizzi through other means.

Motto

Anything in the world is ours to take, and in exchange we offer order.

Beliefs

• Always repay debts.
• All magic and all crime should be conducted under our auspices.

Goals

• Enrich selves.
• Control all forms of magic.

Hooks

• The PCs need a divination, and the only purveyors in town are ’Valizzi.
• Returning to town, the lone guard at the gate doesn't remember them at all. He is being memory-wiped with some regularity by potion smugglers.
• The flood of magic items brought back to town by the PCs has upset a local wizard, who calls in a favor from the ’Valizzi to see about cutting off the supply.
• The PCs perform some minor magical service in town, for example a healing or a detection spell. The ’Valizzi won't be happy to hear about upstarts cutting in on their turf.

The Chuckleheads

1. Father Corx

Overview: Father Corx is a dapper priest who goes way back with the ’Valizzi.

Appearance: He has neat salt-and-pepper hair and a low booming voice.

Abilities: He has the spellcasting ability of a low-level cleric.

Talents: He assuages people's guilt, including his own.

Mannerisms: He takes long pauses between sentences (or as dramatically appropriate).

Interactions: He is very understanding of others, to the point of condescension.

Knowledge: He always knows how to make things right and who to set up meetings with.

Ideals: A strong, if twisted, belief in hierarchy governs his actions.

Bond: He is childhood friends with the current leaders of the ’Valizzi, and sees no conflict between his religious teachings and their actions.

Secrets: He has covered for the ’Valizzi in several large court cases, using his station to sway the jury.

2. Jerry "the Ferret"

Overview: Jerry used to be a smuggler and a burglar with the ’Valizzi.

Appearance: Even bent over a cane, his wiry frame is still tall.

Abilities: He can open a door to any other door in the same building, one-way, provided they both have doorknobs.

Talents: He misdirects at every scale.

Mannerisms: His teeth whistle slightly, enough to stop him casting spells (part of the terms of his release).

Interactions: Jerry is faultlessly polite and fabulously generous, so long as he thinks he might learn something from you.

Knowledge: He can get into and out of any building, or at least knows how.

Ideals: Jerry is a self-made man and values that independence.

Bond: Jerry's only loyalty is to his immediate family, to the chagrin of his former compatriots and cell-mates.

Secrets: Jerry knows where the gold from his last big heist is buried, and intends it to be his son's inheritance.

3. Nicalos, Old Fredward's Boy

Overview: Nicalos is a goon and errand-runner employed by the ’Valizzi.

Appearance: His personal grooming is impeccable, but he wears outdated, fraying clothes.

Abilities: Nicalos is slightly un-magical, and if a spell can fail around him, it will tend to.

Talents: Nicalos has a real knack for hiding and sneaking.

Mannerisms: His speech is heavily-affected cartoonish mob-speak (Nyah, see?).

Interactions: He always seeks to make himself look good, and to win approval from others.

Knowledge: Whatever the characters' business with the ’Valizzi, Nicalos will have a better understanding of the implications and repercussions than they do.

Ideals: Nicalos has aspirations to run the show, and will take initiative given the opportunity.

Bond: Nicalos keeps his mother sheltered from his activities a ways outside town.

Secrets: Depending on the characters' goals, Nicalos' are either exactly the same or exactly opposite. He will not let on to this.

4. Barb, the Lightning Witch

Overview: Barb is an electric witch, an invaluable asset to the ’Valizzi.

Appearance: Short middle-aged woman, long blond hair always floating on end with static charge.

Abilities: Barb can unerringly send one-way messages to the deceased.

Talents: Barb has a flair for the theatrical, which serves her well in séances and rain-calling.

Mannerisms: Cackles.

Interactions: Barb's interactions with others are purely transactional, and she works to maintain her image at all times.

Knowledge: Barb has a finger on the pulse of all the metaphysical movers and shakers, like who is researching what spells, where Excalibur was last seen, which godlings haven't been properly appeased, etc.

Ideals: Barb delights in the exercise of power.

Bond: Everyone owes Barb something, and it makes them uncomfortable, and she knows it.

Secrets: She can no longer charge the batteries that power her magical equipment, and is currently running on fumes.

5. Buddy, the Actual Elected Mayor

Overview: Not a wizard, not a mobster, but in deep with the ’Valizzi regardless.

Appearance: A broad-shouldered, square-chinned, deep-voiced, white-toothed perfect politician.

Abilities: Buddy commands local law enforcement and municipal services.

Talents: People trust Buddy, unless they know him well.

Mannerisms: When he's thinking, Buddy fiddles with a fancy gold fountain pen, occasionally dropping it and surreptitiously retrieving it.

Interactions: Buddy will readily agree to any seemingly-fair deal, but won't do any work to uphold his end of a bargain.

Knowledge: Buddy knows just the right words to sway an audience, or the right gestures to get their attention.

Ideals: Buddy believes in the status quo and in his city. He believes that all of its faults are necessary, and that he can protect it.

Bond: Buddy knows who got him elected.

Secrets: Buddy knows where the bodies are buried.

6. Ruby

Overview: Ruby is a promising young wizard that the ’Valizzi is actively recruiting.

Appearance: She has ankle-length black hair and wears baggy clothing, but she's in there somewhere.

Abilities: Ruby knows a charm to silence her movement at will.

Talents: Ruby learns quick, be careful what spells you let her see.

Mannerisms: When she's not talking, Ruby stands unnervingly still.

Interactions: You can rely on Ruby if she says she'll do something, but she doesn't say much.

Knowledge: She has some idea what the magical abilities of everyone else in the wizard mafia are, and how they work.

Ideals: Ruby values novelty, and will want to try anything new that she hasn't seen before.

Bond: The ’Valizzi bankrolled her magical education, and she's working off that debt one job at a time.

Secrets: Ruby wants out, and will leave once her obligations are fulfilled.

Thoughts

I really like the 5e method of defining an organization, which I first saw used in Krevborna and Cinderheim, so it was a natural fit for defining the organization. Then I thought I'd try the 5e method for defining NPCs also, which ended up being a lot more work than I would normally put into a single NPC.

Things That are Getting Me Through

Everyone is experiencing these times differently. I'm fortunate enough to have a job that I can do remotely and to not be alone in my isolation. But here's some of what's been keeping me going, and I hope it can help.

Pets

I have a cat. I feed her every day. She sits on my lap and purrs and bites me.

We also caught a sourdough starter. I feed it every day. It has not yet bitten me.

Self Care

Because I now live what my father has generously called "the life of a scholar", I don't get a lot of sunlight. I find that vitamin D supplements help me sleep better.

I was never a gym person, but the total lack of activity has started to wear on me. I've taken up the seven-minute workout, which I like much better than I thought I would.

The first few weeks without shaving are all terrible scraggly neck beard for me. With nobody to see it, what better time to push through that period and find out what other terrible scraggly facial hair I can grow?

I miss math and I also miss programming. I've taken the opportunity to write a couple of janky scripts: one that generates thumbnails for video files, and one that removes cruft from CBZ and CBR files, shrinks them a little, and converts CBR to CBZ. Use at your own risk, but I am proud of them.

I brûléed a Cadbury creme egg. It's worth doing at least once.

Media

Before this, we sometimes saw a comedy show called Spoons & Toons & Booze. They've become Spoons & Toons & Booze & Zoom now, and they're raising money for the employees of the theaters where they used to perform.

On a smaller scale, we've hosted a few movie nights with distant friends on twoseven.xyz. It works about as well as any new tool, which is to say, plan at least an extra half-hour at the beginning for socializing and troubleshooting. Test what you're planning to do by yourself first.

On Netflix, after The Great British Bake-Off, Terrace House might be the most bizarre and calming show I've found. Six young Japanese adults live in a nice house and get nice cars. There don't seem to be any stakes or anything? They all keep going to their normal jobs and stuff. After each episode, a bunch of enthusiastic commentators remark on all the drama that may or may not have happened. It's pretty good.

Sylvan Esso released a new live album.

Podcasts

Lots of podcasts will keep you informed or expose you to new and complicated ideas. These are (mostly) not those.

Phoebe Reads a Mystery

Phoebe Judge, host of Criminal, reads a mystery novel one chapter at a time. She started with The Mysterious Affair at Styles, then The Hound of the Baskervilles, and currently The Moonstone.

St Elwick's Neighbourhood Association Newsletter Podcast

Recently launched by Mike Wozniak, frequent guest on The Beef and Dairy Network. The usual arc of an episode is roughly: bizarre to sad to cringe to absurd. It's pivoted into the current situation seamlessly by releasing shorter, more frequent episodes, and I think they're stronger for it.

The Tranquillusionist

Helen Zaltzman of The Allusionist (same feed) reads odd things in a calm voice, with musical accompaniment by her husband, Martin Austwick.

Make My Day

Josh Gondelman has a single guest on to play a made-up game show, where points are given to answers based on how much they cheer him up. Money is given to charity. Strange motivational speeches are delivered.

Seltzer Death Match

Self-explanatory, I think. They've been editing through a backlog recently, so there's a bunch of new ones.

372 Pages We'll Never Get Back

Two of the minds behind RiffTrax read through bad books. My cousin recommended this to me and in exchange I recommended

Bad Books for Bad People

Jack Shear and Tenebrous Kate read through books that they think the other will enjoy/hate.

Monster Man

James Holloway reads through old monster manuals, a couple entries at a time. I particularly enjoyed the episode about mind flayers, and why they are the perfect 70's Doctor Who villain. Nominally topical!

Others

Life is short. Also have these recommendations, with the understanding that I don't enjoy them any less for having lost the energy to describe them:

• Cooking with Paula McIntyre
• Scotland Outdoors
• The Boring Talks

Games

When there was less on our minds, my wife and I would set aside days to solve mysteries, things like Sherlock Holmes Consulting Detective or T.I.M.E. Stories. These days we play a lot of LEGO Batman 3, which is good because neither of us is any good at video games and it's fun regardless.

Quarantine has also gone on long enough that I found my old RuneScape account. While my old character is still sitting there in RS3, I've been playing Old School RuneScape, and it's really interesting to start again at the beginning.

I've started playing in a Lasers & Feelings game with a bunch of doctors and it's good fun and breezy. The same group is also getting ready for a more involved 5e game which should be interesting.

I've been playing in a West End Games/d6 Star Wars game and it's wild. The rules can be found online practically by accident, and they're worth a look.

Finally, it looks like I will be GMing again, likely 5e. I've never actually run 5e proper, but I'm hoping I pick it up easily enough.

7d6, split between two attributes

On 20 Feb 2020, PlanetNiles (Ey/Em) NB on the workshop channel of the OSR Discord asked¹:

7d6, split between two attributes

What values are even possible?

One way to think about this is not in terms of the values themselves, but in terms of the difference between the two paired attributes. The maximum difference is when all dice are allocated to one attribute, leaving the other empty. This is then equivalent to the sum of the dice.

The minimum difference between the two is a form of the well-studied partition problem: given a set of numbers, can it be split into two sub-sets with equal sums? A heuristic method that can usually² find the smallest difference between our two subsets (and therefore if they can be equal or not) is called the Karmarkar-Karp method, which works as follows:

1. Take the two largest numbers in the set. Assume that these will go in opposite partitions.
2. Because they will go in opposite partitions, we subtract them from each other.
3. Instead of now deciding which partitions they will go in, we return the difference between them to the set. This is effectively deferring that decision until later.
4. Repeat steps 1-3 until there are only two numbers left. The difference of these last two numbers is the smallest difference we can make with this set.

Note that we didn't actually find what the partitions are, only what the difference between them is here. I hope this is clear enough, but I'm also sure that clearer explanations can be found with minimal searching.

With some wrangling, we can produce this lovely chart, where a heavier hexagon is a more likely pairing.

Hexagons are elegant, as odd/even combinations that cannot occur are skipped naturally.

From this, it looks like the system gives a lot of flexibility in assigning scores to your attributes.

What Values are Likely?

I have a concern about this mechanic though: if you give someone a range of numbers, and tell them to pick one, they will tend to pick in the middle³. If a player is dead-set on being boring, how boring can they be?

It might not be exactly true, but we can show the distribution of all the high stats and all the low stats together, and then compare them to the distribution of 7d6 literally divided in half (which is approximately normal).

This doesn't look great, but PlanetNiles has actually already got us covered here:

Of course. I'd further consider including subsystems where the difference between attributes had some sort of effect. So favouring one over the other would prove beneficial in some way, or at least open up different options.

The strength of this mechanic then will rely on the strength of the system. Given the range of the first figure though, I have confidence that an interesting system could be built here.

What about other mechanics?

Suppose we were looking for a similar mechanic, except that it would force a difference between the attributes where possible. We might expect intuitively, that fewer dice and larger dice are harder to partition effectively. This table gives the probability of a forced difference (although does not consider the size of that difference).

From this table, I thought I would look at 3d20, because it forces a difference the most often. The figures below have the same interpretations as the similar ones above, but for 3d20 instead of 7d6.

Python

This was my first project using Python, and I think it's an all right language. It'll probably displace Octave in my repertoire, but I'm sure I'll be right back at Perl if I start doing string stuff again. The code's a bit janky, but you can take a look here.

---

1 “Asked” a little more directly this time, if only because I asked first.^back

2 According to Wikipedia, this method is “bad for instances where the numbers are exponential in the size of the set,” so like, probably fine?^back

3 I only know this anecdotally: if you tell a plant operator to keep some process temperature between a high limit and a low limit, without fail they will control it to the middle of the two. It makes sense to a person, but the optimal temperature is provably at one of the two extremes. Possibly, this is an extension of the anchoring effect.^back

Game Jams and More to Fight Social Isolation Blues

I added art to my entry in the Troika! Tarot Jam (11 days left at time of writing), and I now consider it finished. For these I wanted to communicate a lot of flavor without the standard explanatory preamble, to help them fit in such a small space. I think a lot of the backgrounds would be better remixed, but can stand alone as-is. I only hope that the illustrations aren't so specific that they interfere with anyone's interpretations of their own characters.

I don't technically have more time during social isolation because of work. But I still need things to occupy my mind, so here are some RPG-related activities:

• Troika! Tarot Jam
• Good Morning Apocalypse Game Jam
• The Great Khan's Celtic Adventure Contest
• Cone of Negative Energy: Please Stay Home Bundle
• Free PDFs from Exalted Funeral

There are also lots of other things:

• Entertaining Yourself in Times of Quarantine
• Virtual School Activities (all kinds of cameras and virtual museums)
• The National Mall [Cherry Blossom] Bloom Cam
• The Localhost Gallery (A Minecraft Art Gallery)
• Streaming Concerts
• A Streaming Variety Show
• Catch a Sourdough Starter

Please, stay safe and support each other.

The Bees Below

I promised my players I would “concoct some suitable fable” about atomic bees. It's not great prose, but I think it's functional. I didn't find a place to put laser-eyes in the story though, which I regret slightly.

One spring, when Bear awoke, his mother was dead. Unsure what to do, he went to Mole’s house to ask how he could get his mother back, for Mole was very wise.

Mole said, “In my deep journeys, I have encountered rare bees, and you must seek them. Ask them for their honey, which can work miracles.”

Bear traveled through the wood, by paths known only to animals. But in the distance, a hunter saw him, and was afraid.

Bear came to a cave entrance he had never seen before, and descended. He walked for days and nights until he came to a rickety bridge across a deep chasm. Bear was afraid, for he did not know if the bridge could hold his weight. But running across, Bear arrived at the far side, the bridge collapsing behind him.

Continuing, Bear came to a cavern piled with glittering wealth: rubies the size of sunflowers, gold coins from nameless empires, and flatware of the finest aluminum. But Bear, his heart still heavy with grief, cared not for the riches therein. He stopped, and wept “Oh! That such riches could bring back to me my mother! But alas, I must continue.”

Finally, Bear arrived at a meadow, so deep beneath the earth that the sun could not have suspected it to be possible. The flowers there were wrought of precious metals and the grass crinkled gently in the breeze. Glowing gently in the distance was the hive of the atomic bees.

The bees buzzed, “Bear, why are you so far from home, where the only light is the glow of our hive?” “My mother is dead and I cannot go on without her,” Bear wept. “Feed her some of our honey, and it will bring life to her bones,” the bees buzzed, and they gave him a thick honeycomb to bring back with him.

When Bear returned to the cave of treasures, he froze. A fearsome serpent had made its home there and was presently asleep, coiled about a velvet throne. Bear waited outside the cavern, but was impatient. Unsure what to do, he tasted a drop of the honey. His fur stood on end like a frightened cat’s tail, and when he next looked for his reflection in the polished surface of a silver mirror, he could not find it. No longer fearing the serpent, he strode boldly into the chamber. The clinking of coins woke the serpent, but its cries of “who goes there?” and “what have you brought me for tribute?” went unanswered, for Bear had already passed through the chamber.

At last, he came to the chasm, but the bridge was still broken, and the rushing river below could barely be heard so long was the fall. He sat and he wept, for he had come so far, but could not complete his quest. After a while, he grew hungry, and tasted another drop of the honey. At this taste, his teeth hummed like the bees below, and he lifted off the ground. Swimming through the air he came to the other side of the chasm and continued on his way.

Climbing the passage for days and nights, Bear saw the sun in the distance and rejoiced, “soon I will see my mother again!”, and bounded into the wood. But in his haste, Bear did not see the the hunter had set a cruel trap for him, and his legs were taken off by jaws of iron. Bear cried out, “How cruel! To see the sun again, but nevermore my mother!” Remembering the miraculous honey, he tasted another drop. His whole body felt as though it were on fire, and in panic, Bear patted down his limbs. Now he understood that his limbs had grown back, and were as strong as they had ever been.

Rejoicing, Bear walked the hidden paths back to his home, but when he attempted to place a drop of the honey in his mother’s mouth, he found that none was left. At this, Bear wept, and cursed the bees below. Their gift had meant nothing to him. And he cursed so loudly that the bees below heard his curse and were saddened. The bees could share no more, or they would have none for themselves. So they resolved to never again share their honey and to defend it from all others, to spare us all the Bear’s pain.

5d6 but only count straights and matching

On 7 Feb 2020, diregrizzlybear on the GLOG channel of the OSR Discord asked¹:

5d6 but only count straights and matching.

One solution might be to list all the rolls and score each one. This is probably feasible with a script. Instead, I enumerated the “hands”, and then found the probabilities of each of those.

Hands with No Degrees of Freedom

Run of Five

There are only two runs of five: ⚀⚁⚂⚃⚄ and ⚁⚂⚃⚄⚅. There is only one way to “make” each of these hands (“Count”), but because each die has a different face, there are 5!=120 possible orderings of each hand (Permutations).

Hand	Score	Count	Permutations	Odds
⚀⚁⚂⚃⚄	15	1	120	120
⚁⚂⚃⚄⚅	20	1	120	120

Quintuple

There are six possible quintuples, and again, there is only one way to construct each one. While there are 5! possible orderings of five dice, because five of them are interchangeable, there is only one possible ordering of a quintuple (5!/5!=1), which makes a quintuple much less likely than a run of five.

If this seems counter-intuitive, consider rolling one die five times in order. If your first roll is a ⚀, to eventually score quintuples, the next roll must also be a ⚀ (1/6 odds). To eventually score a run of five, the next roll must only be not ⚅ or ⚀ (4/6 odds).

Hand	Score	Count	Permutations	Odds
⚀⚀⚀⚀⚀	5	1	1	1
⚁⚁⚁⚁⚁	10	1	1	1
⚂⚂⚂⚂⚂	15	1	1	1
⚃⚃⚃⚃⚃	20	1	1	1
⚄⚄⚄⚄⚄	25	1	1	1
⚅⚅⚅⚅⚅	30	1	1	1

Run of Three + Double

There are 24 ways to score a run of three + double: 4 runs of three and 6 doubles. Depending on the doubled number, it may be possible to score this as other hands (run of four, triple), but this is never advantageous.

Because of the doubled number, there will be fewer ways to order this hand than a run of five, but more than a quintuple. If the doubled number is in the run, there are 5!/3!=20 possible orderings, and if it is not, then there are 5!2!=60.

Hand	Score	Count	Permutations	Odds
⚀⚁⚂⚀⚀	8	1	20	20
⚀⚁⚂⚁⚁	10	1	20	20
⚀⚁⚂⚂⚂	12	1	20	20
⚀⚁⚂⚃⚃	14	1	60	60
⚀⚁⚂⚄⚄	16	1	60	60
⚀⚁⚂⚅⚅	18	1	60	60
⚁⚂⚃⚀⚀	11	1	60	60
⚁⚂⚃⚁⚁	13	1	20	20
⚁⚂⚃⚂⚂	15	1	20	20
⚁⚂⚃⚃⚃	17	1	20	20
⚁⚂⚃⚄⚄	19	1	60	60
⚁⚂⚃⚅⚅	21	1	60	60
⚂⚃⚄⚀⚀	14	1	60	60
⚂⚃⚄⚁⚁	16	1	60	60
⚂⚃⚄⚂⚂	18	1	20	20
⚂⚃⚄⚃⚃	20	1	20	20
⚂⚃⚄⚄⚄	22	1	20	20
⚂⚃⚄⚅⚅	24	1	60	60
⚃⚄⚅⚀⚀	17	1	60	60
⚃⚄⚅⚁⚁	19	1	60	60
⚃⚄⚅⚂⚂	21	1	60	60
⚃⚄⚅⚃⚃	23	1	20	20
⚃⚄⚅⚄⚄	25	1	20	20
⚃⚄⚅⚅⚅	27	1	20	20

Triple + Double

There are 30 ways to score a triple + double: 6 ways to score one and then 5 remaining ways to score the other (to exclude quintuples, which are already accounted for). As with run of three + double, we must account for duplicated numbers when counting orderings. There are then 5!/(3!*2!)=10 permutations of each.

Hand	Score	Count	Permutations	Odds
⚀⚀⚀⚁⚁	7	1	10	10
⚀⚀⚀⚂⚂	9	1	10	10
⚀⚀⚀⚃⚃	11	1	10	10
⚀⚀⚀⚄⚄	13	1	10	10
⚀⚀⚀⚅⚅	15	1	10	10
⚁⚁⚁⚀⚀	8	1	10	10
⚁⚁⚁⚂⚂	12	1	10	10
⚁⚁⚁⚃⚃	14	1	10	10
⚁⚁⚁⚄⚄	16	1	10	10
⚁⚁⚁⚅⚅	18	1	10	10
⚂⚂⚂⚀⚀	11	1	10	10
⚂⚂⚂⚁⚁	13	1	10	10
⚂⚂⚂⚃⚃	17	1	10	10
⚂⚂⚂⚄⚄	19	1	10	10
⚂⚂⚂⚅⚅	21	1	10	10
⚃⚃⚃⚀⚀	14	1	10	10
⚃⚃⚃⚁⚁	16	1	10	10
⚃⚃⚃⚂⚂	18	1	10	10
⚃⚃⚃⚄⚄	22	1	10	10
⚃⚃⚃⚅⚅	24	1	10	10
⚄⚄⚄⚀⚀	17	1	10	10
⚄⚄⚄⚁⚁	19	1	10	10
⚄⚄⚄⚂⚂	21	1	10	10
⚄⚄⚄⚃⚃	23	1	10	10
⚄⚄⚄⚅⚅	27	1	10	10
⚅⚅⚅⚀⚀	20	1	10	10
⚅⚅⚅⚁⚁	22	1	10	10
⚅⚅⚅⚂⚂	24	1	10	10
⚅⚅⚅⚃⚃	26	1	10	10
⚅⚅⚅⚄⚄	28	1	10	10

Hands with One Degree of Freedom

Run of Four

There are three possible runs of four: ⚀⚁⚂⚃x, ⚁⚂⚃⚄x, ⚂⚃⚄⚅x, where x is our “unfixed” die (our degree of freedom). If x is equal to either the highest or lowest element of the run, then we instead have a run of three + double. If it is equal to a number after either end of the run, then we instead have a run of five. So for ⚀⚁⚂⚃x and ⚂⚃⚄⚅x, x has three possible values, and for ⚁⚂⚃⚄x, x has two possible values. We will also consider the cases where x is “inside” the run and “outside” the run separately, as the number of permutations is different.

Hand	Score	Count	Permutations	Odds
⚀⚁⚂⚃x;x∈{⚁,⚂}	10	2	60	120
⚀⚁⚂⚃x;x=⚅	10	1	120	120
⚁⚂⚃⚄x;x∈{⚂,⚃}	14	2	60	120
⚂⚃⚄⚅x;x∈{⚃,⚄}	18	2	60	120
⚂⚃⚄⚅x;x=1	18	1	120	120

Quadruple

There are 6 possible quadruples, with 5 ways to construct each one (again, to exclude quintuples). There are 5!/4!=5 permutations of a quadruple.

Hand	Score	Count	Permutations	Odds
⚀⚀⚀⚀x	4	5	5	25
⚁⚁⚁⚁x	8	5	5	25
⚂⚂⚂⚂x	12	5	5	25
⚃⚃⚃⚃x	16	5	5	25
⚄⚄⚄⚄x	20	5	5	25
⚅⚅⚅⚅x	24	5	5	25

Two Doubles

There are 15 ways to score two doubles (half as many as triple + double, because it doesn't matter which number is the first multiple and which number is the second). The unfixed die (x) can take any of the four remaining values². A hand of two doubles has 120!/(2!*2!)=30 permutations.

Hand	Score	Count	Permutations	Odds
⚀⚀⚁⚁x	6	4	30	120
⚀⚀⚂⚂x	8	4	30	120
⚀⚀⚃⚃x	10	4	30	120
⚀⚀⚄⚄x	12	4	30	120
⚀⚀⚅⚅x	14	4	30	120
⚁⚁⚂⚂x	10	4	30	120
⚁⚁⚃⚃x	12	4	30	120
⚁⚁⚄⚄x	14	4	30	120
⚁⚁⚅⚅x	16	4	30	120
⚂⚂⚃⚃x	14	4	30	120
⚂⚂⚄⚄x	16	4	30	120
⚂⚂⚅⚅x	18	4	30	120
⚃⚃⚄⚄x	18	4	30	120
⚃⚃⚅⚅x	20	4	30	120
⚄⚄⚅⚅x	22	4	30	120

Hands with Two Degrees of Freedom

Run of Three

There are 4 runs of three: ⚀⚁⚂xy, ⚁⚂⚃xy, ⚂⚃⚄xy, ⚃⚄⚅xy. However, x cannot equal y (else we have run of three + doubles), x and y cannot both equal numbers in the run (else we have two doubles), and neither of x and y can equal a fourth part in the run (else we have a run of four).

For a run of three with no duplicates (for example, ⚀⚁⚂⚄⚅), there are 5!=120 permutations. For a run of three with one duplicate, there are 5!/2!=60 permutations.

Hand	Score	Count	Permutations	Odds
⚀⚁⚂xy;x∈{⚀⚁⚂},y∈{⚄⚅}	6	6	60	360
⚀⚁⚂xy;(x,y)=(⚄,⚅)	6	1	120	120
⚁⚂⚃xy;x∈{⚁⚂⚃},y=⚅	9	3	60	180
⚂⚃⚄xy;x∈{⚂⚃⚄},y=⚀	12	3	60	180
⚃⚄⚅xy;x∈{⚃⚄⚅},y∈{⚀⚁}	15	6	60	360
⚃⚄⚅xy;(x,y)=(⚀,⚁)	15	1	120	120

Triple

There are six possible triples, each with two degrees of freedom (x,y). x cannot equal y, neither of x and y can equal the tripled number, and x and y cannot form a run of three with the tripled number. There are then (5²-5)/2-R=10-R ways to make each triple, where R is the number of runs of three containing the tripled number.

Hand	Score	Count	Permutations	Odds
⚀⚀⚀xy	3	9	20	180
⚁⚁⚁xy	6	8	20	160
⚂⚂⚂xy	9	7	20	140
⚃⚃⚃xy	12	7	20	140
⚄⚄⚄xy	15	8	20	160
⚅⚅⚅xy	18	9	20	180

Hands with Three Degrees of Freedom

Doubles

There are six possible doubles, each with three degrees of freedom (x,y,z). None of x, y, and z can equal each other, none of x, y, and z can equal the doubled number, and x, y, and z cannot form a run with the tripled number. There are then 5!/(3!*(5-3)!)-R1-R2 =10-R1-R2 ways to make each double, where R1 is the number of runs of three (4) and R2 is the number of runs of four containing the doubled number.

Hand	Score	Count	Permutations	Odds
⚀⚀xyz	2	5	60	300
⚁⚁xyz	4	4	60	240
⚂⚂xyz	6	3	60	180
⚃⚃xyz	8	3	60	180
⚄⚄xyz	10	4	60	240
⚅⚅xyz	12	5	60	300

Other Hands

Other hands are not possible with 5 dice, but I did not bother to prove this more formally. Instead, I can show that all hands are accounted for: there are 6^5=7776 possible rolls (in order), and the sum of all the “Odds” of the above hands is 7776.

Results

Now we can sum the odds by score (instead of by hand) and normalize them. This gives us the following distribution.

Score	Odds
1	0
2	300
3	180
4	265
5	1
6	940
7	10
8	355
9	330
10	741
11	80
12	915
13	40
14	620
15	791
16	405
17	100
18	760
19	140
20	296
21	140
22	160
23	30
24	105
25	21
26	10
27	30
28	10
29	0
30	1

The minimum score is 2, maximum 30, mean ~12.4, median 11, and mode 6. My spreadsheet is a bit messy, but you can see it here. Let me know if anything here seems off.

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1 “Asked” is a strong word. Nobody asked for this.^back

2 In the case of ⚀⚀⚁⚁⚂, both run of three and two doubles would score 6. For convenience, we will consider it as two doubles, because restrictions to exclude it are already part of the math for a run of three.^back