Mazes & Mages


Dedicated with love to my great grandmother Rosa “Southern Omi” Carniol Strulovici

In mathematics the art of asking questions is more valuable than solving problems.

– Georg Cantor

I believe that even a smattering of such findings in modern science and mathematics is far more compelling and exciting than most of the doctrines of pseudoscience, whose practitioners were condemned as early as the fifth century B.C. by the Ionian philosopher Heraclitus as “nightwalkers, magicians, priests of Bacchus, priestesses of the wine-vat, mystery-mongers.” But science is more intricate and subtle, reveals a much richer universe, and powerfully evokes our sense of wonder. And it has the additional and important virtue—to whatever extent the word has any meaning—of being true.

– Carl Sagan (Broca’s Brain)

To the teenager:
In sword and sorcery fantasy games like Dungeons & Dragons, the outcome of everything you do is based on luck – you roll the dice (or the computer randomly decide) and this determines whether you succeed or not. How cool will it be if the same game was instead based on skill? Then you control your own fate, you face a real mental challenge, so it’s more satisfying once you solve it!

To the parent:
Just like our grandparents’ generation used slide rule, and our parents used a basic calculator, and we used a scientific calculator, so will the next generation use mathematical apps. Why not give them a head start in an enjoyable and exciting way? Wolfram made cool and affordable “Course Assistant” apps that go on your kid’s mobile device. This book acts as the “Dungeon Master” and asks the brave adventurer to solves puzzles using the apps, and afterwards the book gives a simple everyday explanation so no one is left behind!

Preface

This book teaches Wolfram Course Assistant Apps (mathematics study software), using Dungeons and Dragons (fantasy role-playing game).
At first glance, these two subjects seem to be at odds with one another: Math* and fantasy?!? Study and game?!?
(Please note: throwing dices and adding numbers is not math, it’s elementary school arithmetic; What’s more, when these games move to the computer these calculations are done for you by the computer, so this means they are not a real part of the fun of the game, only a technical necessity).
The most fruitful discussion of these two together that I found is in the DnD forum of Reddit:

When should high-level math pop up in game?
https://www.reddit.com/r/DnD/comments/7s7ub9/when_should_highlevel_math_pop_up_in_game/

So roughly speaking, this person is a Dungeon Master (the “story-teller”) of his D&D playing group and he’s asking the forum: nerds have fun with D&D; nerds have fun with math; why not combine the two?

Put this way it sounds reasonable, but still, most of the answers are completely negative (don’t do it – it will ruin the game), or somewhat negative (you go ahead – but I will never do it). Why is that?

1st mistake: NERDS
Only few people are good with BOTH sides of the brains. Most “nerds” are good with EITHER the left side OR the right side.

Lateralization of brain function
https://en.wikipedia.org/wiki/Lateralization_of_brain_function

The left side of the brain handles MATH:
analytical thought, detail oriented perception, ordered sequencing, rational thought, verbal, cautious, planning, Math/Science, logic.

The right side of the brain handles D&D:
intuitive thought, holistic perception, random sequencing, emotional thought, non-verbal, adventurous, impulse, creative writing/art, imagination.

Society categorizes both kinds as “nerds” because both kinds are smart and intellectual, but really there are two different kind of “nerds”. It’s only TV shows like “The Big Bang Theory”) that display fake and inconsistent “nerds” the supposedly do all things “nerd”. The show shows people work on quantum physics and then find great interest in comics and cosplay. Nonsense!

2nd mistake: FUN
https://en.wikipedia.org/wiki/Educational_entertainment

The whole concept of edutainment along with the implied message that you must make them laugh otherwise they will not listen is a crime against intelligence.
This is heavily discussed in my first book, so I’ll just remind the bottom line: real learning IS fun! You do NOT need to disguise the math, you just need to motivate it so that the student WANTS to learn.
As kids we are all-natural scientists and only when we reach school that they ruin this for us by force feeding. Kids who had the good fortune to have a parent (or a special teacher) who nurtured their natural curiosity about science (and NOT shove science down their throat) think science is very cool and fun, and indeed grow up to be scientists and engineers.
(I’m treating math and science interchangeably here because math is almost always inspired by real life science and it’s almost impossible to find math that is not useful somewhere in the real world).

3rd mistake: MATH

We all know what math is… or do we?
We all know what “doing math” means today (even if we ourselves can’t do it). This is what the people rambled about in the forum. But what will “doing math” mean tomorrow?
In the next few minutes I will explain about “Wolfram Mathematica” and “Wolfram Alpha”:
https://en.wikipedia.org/wiki/WolframAlpha
and I expect two questions to bother you:
(1) Surely math is a lot bigger than any single product – is it really that important?

(2) Surely there are other similar products – why the fixation on this specific one?

The answer is that you can do all these things with other software, but with Wolfram’s tools it’s a lot integrated which translates to easier and friendlier. For example, in Wolfram Mathematica, you can write one line of code to plot a graph, but then you can add the command “manipulate” at the beginning of the line and the “frozen” graph transforms into an interactive graph where you can slide with your mouse the slide bar and immediately how it changes!
This is UNIQE to Wolfram Mathematica – no other software can do it!
You can see amazing examples of this power in Wolfram’s Demonstrations:
https://demonstrations.wolfram.com/

So in Mathematica you need special syntax.

In Wolfram Alpha you can ask in natural language anything without any special syntax, and it does the calculation for you! It’s somewhat like Google but instead of the raw information in Google, Wolfram Alpha “chews” the information for us and bring us the result of any calculation we asked for!

For example, I wanted to know: How much time it will take me to reach Mars?

I asked: how far is mars
Alpha answered: Mars | distance from Earth: 228.2 million miles
I asked: speed of fastest space ship
Alpha answered: fastest recorded Earth-relative speed of a manned space vehicle: 24790 mph
I asked: 228.2 million miles / 24790 mph
Alpha answered: 383.6 days

This sounds way too much, but remember Wolfram|Alpha calculated the distance for RIGHT NOW. It also gives us a small graph that shows the distance between Earth and Mars and we were 5 times closer 9 months ago. Also we asked about the max speed, when in reality we waste time in accelerating and decelerating. Also we will probably send a bigger ship with more people. But you get the idea!

OK so Wolfram|Alpha isn’t perfect, we have to ask the right question.

Another imperfection of Wolfram|Alpha is that it cowers away from “dangerous” calculations, for example, I tried to find out how much money do the world’s billionaires have, or how much the 99% of the population have, in all sorts of phrasing and got nothing, even though I’m sure it’s very interesting for a lot of people.

On the other hand, typing “total wealth of 99%” in Google, immediately brings this article from the BBC:

Oxfam says wealth of richest 1% equal to other 99%
https://www.bbc.com/news/business-35339475

Oxfam also calculated that the richest 62 people in the world had as much wealth as the poorest half of the global population… [from the graph in the article you understand that now it’s much worst, the rich get richer and everybody else gets poorer. this article is pretty old, it’s from 2016!]

But that’s just me whining about minor imperfections in what is no doubt the most friendly and powerful (in the sense of what the average person and the average scientist / mathematician can do with it) software EVER! I guess the very top of the top experts in the world, have access to huge dedicated super computers and they optimize all their code manually and squeeze a little more power from their machine. So they don’t need the automation and the “write once, run anywhere” simplicity. But for all the rest of us – this is the closest thing to what “The Hitchhiker’s Guide to the Galaxy” talked about!
https://en.wikipedia.org/wiki/Phrases_from_The_Hitchhiker%27s_Guide_to_the_Galaxy#Answer_to_the_Ultimate_Question_of_Life,the_Universe,_and_Everything(42)

By the way, one of the people of the above forum thread said he/she uses Wolfram|Alpha for questions like: “How much does a 5 ft cube of stone weigh?”.

OK. So do we use Wolfram Alpha or Wolfram Mathematica in this book? Yes we will use this great power but it’s hidden “under the hood”. The Course Assistant apps
https://products.wolframalpha.com/courseassistants/
have a much simpler interface. Alas, the price comes in flexibility: In the apps we don’t have any syntax, but we can only ask specific “questions”.

This is a good thing because we want the pupil (sorry… our brave adventurer!) to have very constructed method to approach each new subject. So the fact that our brave adventurer has just (for example) two textboxes (“field”, a place where the user is expected to enter a single item of data) on the screen is actually good for learning something new the first time around.

Summary
In this book we avoid these 3 mistakes:
• The “target market” are people who are interested in math (or wish to learn math).
• This book teaches math, unapologetically. The fun will come from learning new things.
• To keep the momentum and avoid distraction the apps do the calculations for us.
Hence, we in this “lab experiment” (the reader Vs. the book), we will be able to teach math through D&D ! But can this experiment be successful in the field (adventurers Vs. the Dungeon Master)?

Mazes & Mages game?

I guess the next question to arise now is can we make it into a game, let’s say M&M a mathematical fantasy game, where instead of being armed with those “hidden-variable” dice [thank you very much to Diacritica from Wikipedia for the photo! ] ,

Each player will be armed with “cause-and-effect” apps:
[ this is a screen capture from my own smartphone and you can see many of the Wolfram Course Assistant apps. ]

Side Note 1:
Just to clarify: I don’t get ANY money or ANY benefit from Wolfram in ANY way, and also these apps ($3 for each app I think) were bought by me a few years ago out of curiosity. I am using these apps because I like them! NOT because I’m getting anything from this!
Side Note 2:
Did you notice how beautiful the shape of the D&D dice is? With one exception (which we’ll get to in a moment) they are the very special Platonic solids that were so admired by the Greeks that they thought the universe is built from them.
https://en.wikipedia.org/wiki/Platonic_solid
They are the ONLY “air-tight” solids you can build by “tiling” the same regular polygon (like equilateral triangle). Throughout history they were chosen for dice because they can be made fair because of their great symmetry.
Did you guess which one is the exception? The dice with 10 sides doesn’t belong! it’s only used in D&D (even though it’s NOT a Platonic solid) because we humans are used the to the number 10 so much (because we happen to have 10 fingers so we are used to a 10 based system). Like if we want to have a number between 1 and 100 for percent it’s easy to cast (throw) two 10 sided dice.

So I’m returning to the question: assuming we can make an enjoyable book, can we make Mages & Mazes into a game that people actually can play in an enjoyable way?

Before I start to answer, I have to confess:
I NEVER PLAYED D&D IN MY LIFE!
For me it all began when I saw this AMAZING cover picture by fantasy illustrator Larry Elmore (the picture is called “Basic D&D Ancient Red”). Of course all rights reserved to these people and companies:
Larry Elmore
https://en.wikipedia.org/wiki/Larry_Elmore
TSR
https://en.wikipedia.org/wiki/TSR,_Inc.
Wizards of the Coast
https://en.wikipedia.org/wiki/Wizards_of_the_Coast
Many thanks to Tom Simpson for putting this picture in his Flickr account
https://www.flickr.com/photos/randar/
https://www.flickr.com/photos/randar/30858273483

Can anyone look at this painting and NOT yearn to play D&D?
So D&D came to my attention when I was about 11 years old, and so my mother bought me an adventure module
https://en.wikipedia.org/wiki/The_Veiled_Society
And if I remember correctly she agreed that if after playing this I will want to get more serious, then she will buy me the more pricey set of rulebooks (the above picture is the cover of one of these books)
https://en.wikipedia.org/wiki/Dungeons_%26_Dragons_Basic_Set#1983_revision

So I arranged with smart friends from my class at primary school (Benny Cohen, Avichai Chacham and Itamar Freilich I think) to meet up at Benny’s home and play (I guess I was supposed to be the DM), but another friend who was not dumb but more of a brute (Ran Nakash)
https://en.wikipedia.org/wiki/Ran_Nakash
Also asked to come to and we agreed, but once we started he deliberately ruined the whole session. He wanted to be a cleric or he didn’t, I don’t remember what happened. Anyway that was the first and last time I tried to actually play D&D.

So I guess you’re thinking: Then why am I writing a book about something I don’t know anything about? The truth is my motivation was to learn Wolfram Mathematica, and as a stepping stone to learn the Course Assistant apps myself. So since for me the best way to learn is to teach:
Docendo discimus – “by teaching, we learn.”
https://en.wikipedia.org/wiki/Docendo_discimus
otherwise I’m too lazy to do anything, so I decided to find a friendly way to teach the Course Assistant apps to the general public. So my first instinct was to combine it with stories of either science fiction like Douglas Adams books (The Hitchhiker’s Guide to the Galaxy) or fantasy like Terry Pratchett books (Discworld). But it had no connection to real life, so for me it lacked interest. Like there is a book called Octavo (Latin for eighth) so you HAVE TO do something with 8 for no reason at all. Or there is an answer with 42 which I mentioned earlier, so the same goes with 42. You see? Boring.
https://en.wikipedia.org/wiki/Unseen_University#Octavo

A long time ago I forced myself to read the first trilogy of Dragon Lance:
https://en.wikipedia.org/wiki/Dragonlance_Chronicles
So watching the film adaptation of Discworld made me think that if I take something A LOT meatier then I will have enough material to work with and still make it interesting. So for me D&D does to fantasy what Mathematica does to mathematics, it consumes everything and let’s you work with anything you choose.
So it’s not like I don’t like fantasy, I enjoyed The Hobbit (book not film), I enjoyed The Lord of the Rings (films not book), I was forced by an ex-girlfriend through the whole Harry Potter (films not books), and I was a child in the 80’s so I LOVE all the stuff like The Princess Bride, The NeverEnding Story, and I read Dracula 3 times, and also once the “sequel” book The Historian which was meh. Oh and one of my top favorite movies of all time is The Flight of Dragons
https://en.wikipedia.org/wiki/The_Flight_of_Dragons

Also my favorite computer game ever is Quest For Glory 4, which is set in the distant land of Mordavia, a world full of undead that is “a mix of Slavic folklore and Lovecraftian horror”.
https://en.wikipedia.org/wiki/Quest_for_Glory:_Shadows_of_Darkness
I would come from high school and play this thing for hours and hours and hours.
I even downloaded and tried to do the “tutorial” mission in World of Warcraft and tried (in vain) to understand why it’s such a big thing.
So you see I’m not all bad, it’s just that I have a deep need for what’s REAL, and I’m sure you will notice this in this book.

But for the purpose of writing this book I dedicated about one hour’s worth, to the beginning of each of the following activities:
• watching Critical Role (first campaign) on Youtube
• playing Baldur’s Gate (enhanced edition)
• reading the core book Player’s Handbook
• reading famous module The Mud Sorcerer’s Tomb
So judging by this small taste of the D&D world, what might prevent M&M from becoming an actual game?

1st problem: can’t appeal to masses
From what I understand, in D&D the main thing is the social element:
In real life (Critical Role) they laugh at the Scottish accent of the DM, they laugh if the half-orc said something silly, they barter on the price of things like the cost of the inn. Basi
cally they enjoy doing together the everyday things that they do everyday masked in a thin vail of “antiquity”, things that they can all take part of because they all understand very well. In virtual life (World of Warcraft) I heard they meet with people they already know in the game and sit by the camp-fire and drink beer and tell stories. So again the “charm” is the common activity that everyone can take equal part in either as active actors or passive listeners. The fantasy setting is just an excuse for the social interaction.
This is NOT the case in M&M, where the main thing is solving math problems:
Here the fantasy setting is just an excuse for analyzing problems in a scientific way and solving them elegantly with math.
So I want to be very clear: the problem here is not that a high level of math is needed. The DM can always “throw a bone” so to speak in the form of releasing more and more clues to the group of adventurers about the current problem. So the problem is not the difficulty. The problem is the uneven level of the players. The problem is not that everybody is not up to par; that can be fixed with clues and help. The problem some are good and some are bad. You can get a taste of this in today’s gameplay if one player has finds it difficult to make the calculations. It’s embarrassing for him and frustrating for the group. So in M&M this will get much worst.
So we really want to avoid this scenario of some find it very easy and some find it very hard.
How can this problem be solved?
In other games, let’s say basketball, maybe you can’t do what the best player on the team just did, but you can certainly understand it, and you can help in your own small way, like assisting or blocking a rival, and you feel part of the team. As a spectator sport, if you watch the NBA, although you couldn’t grasp the actual ball, you can still grasp the idea of what they did (at least when it’s played in slow motion). Everybody can get and admire what the best player just did. In M&M most people will require a lot of explanations.

One possible solution will be to play in groups that are more homogeneous, like when the math wiz of the class, who thinks he or she is the only one interested in math, and suddenly they go to a training camp for a math contest and they meet a lot of teenagers like them and it’s a delightful experience for everybody. So although it may sound very formal to make “entrance exams” for classifying players in a game into level categories, if we did that I guarantee they will have a lot more fun! If we accept different levels in chess for example, why not in M&M ? So that’s solvable.
Another possible solution is to slow down the game A LOT and provide help. In chess it’s called correspondence chess, like we see in movies when two old friends make one move on the chess board each day like in the very first episode (pilot) of Macgyver for example.
https://en.wikipedia.org/wiki/Correspondence_chess
So perhaps we can make a platform for teenagers to do one “move” per day, or week or whatever, like at the end of math class, and every time the teacher will publish the full explanation before the next time, and give them the opportunity to ask privately for things they don’t understand, like in a private email or phone so they’re not embarrassed by anyone. If some of the parents are interested, they can also be “certified” as teaching assistants.
https://en.wikipedia.org/wiki/Teaching_assistant
The following day the teacher can solve it on the blackboard in class (not asking the class, so it doesn’t become a competition with frustration/embarrassment) before giving the next riddle.
Coming to think of it I turned it again from a game into a book thing, but I think it’s still a major improvement over the dry presentation of the material in a typical math class!

2nd problem: can’t be mechanized

When I say mechanized I mean not just literally taking the campaign setting of Forgotten Realms
https://en.wikipedia.org/wiki/Forgotten_Realms
and turning it into the video games Neverwinter Nights
https://en.wikipedia.org/wiki/Neverwinter_Nights
You can see the gameplay here thanks to Lacry on Youtube:
Neverwinter Nights: Enhanced Edition Walkthrough Gameplay Part 1 – No Commentary (PC) (NWN: EE)
by Lacry
https://www.youtube.com/watch?v=zY3h78O_NJ4
for example in all these D&D games the battle is simplified and standardized, which is the exact opposite of what we are trying to do in M&M !
So what I really mean is that there is no algorithm for writing the M&M game.
A D&D Dungeon Master (DM) doesn’t need to think about the mechanics of the battle. He or she just pick the monster from the Monster Manual,
https://en.wikipedia.org/wiki/Monster_Manual
and while the design of the monsters have a lot of imagination in their design, it really doesn’t matter how it works exactly because it translates to simple standardized terms like “fire damage” or something like that. In M&M the exact details of how it works matter a lot and make all the difference between life and death!
In the game of D&D it’s not that much harder to be the DM compared to being a player. True, the DM does the “homework” of preparing the adventure, writing the story (if it’s not a ready made adventure) but it’s mostly about keeping things coherent in terms of design (like be faithful to the mythology he/she chose), and balancing the difficulty by giving some help to the players during gameplay. So more or less the DM is picking ready made building blocks and more or less randomly tossing them together.

On the other hand the players at least in the micro scale have a lot of room for creativity, like a friend (Yonatan Lev) once told us that he played as an evil wizard so he cast a sleeping spell on the his group and ran off with all their money. So although I never played, if it’s anything like real life, most people use what talent they have to successfully do very banal things, like conspiring with another person to trick the third person, etc. Most people never really invent anything original.

But in M&M that’s exactly the requirement from the DM. He or she has to come up with very original contraptions that are tailored specifically for the current situation, and break the mold every time! This is so much harder then what the player has to do.
I recommend that if you can’t implement the mechanisms for balancing the “scientific thinking” ability gaps between the players that we talked about, then at least separate the group according to these abilities, and make the upper half (the ones who are able to invent original stuff) to be the Dungeon Masters and make the lower half (the ones who are only able to use ready made stuff) to be the players.
So since there is NO algorithm for making M&M adventures, I’ll tell you my pseudo algorithm: I take the name of the “function” (for example: GCD) and write in Google that name, blank, intuitive explanation or real life example, and so on, and most times Google gives you links to forums like Stack Exchange Mathematics, where someone already asked this and very smart people answer him/her.
https://en.wikipedia.org/wiki/Stack_Exchange
In some cases you can also find examples in Wikipedia, for example physical principles based on that mathematical “function”.

3rd problem: can’t use teamwork

As I discussed in my previous books especially “What’s Wrong With Mathematics and how to fix it”, when it comes to actually inventing something original there is no such thing as teamwork. That’s why you see basketball played by a group together, but you don’t see chess played by a group together. If a chess grandmaster needs to think about what will happen several moves ahead, there is no way he can communicate this to another player (even an expert player) in real time. Probably he can’t lay it into words at all, it’s more like a bunch of intuitions that are based of a lot of experience from the past in similar situations.
So of course, I’m not comparing the problems in this book to the sophistication of chess, but it’s the same kind of problem: the fragments of ideas that float in your head are not tangible enough to communicate to another person until you put the puzzle together in your head. And once you did all the hard work yourself, and the other person can only compare it to his/her complete solution which they did by themselves, and you just pick the better solution. There is no real teamwork.
Teamwork can happen only where each member of the team has a very well defined job to do, and in creative thinking is not well defined at all. You can’t split the task of coming up with an invention into sub-tasks and then put their products back together into a complete whole. You can only do this when you already know what will be the final outcome, but then it’s just implementing a technique that you know well to a very similar situation to what you’re already used to.
On the other hand in D&D anything goes, because nothing has to stand to the test of reality. There is a German proverb: “paper is patient” which means you can write anything and the paper itself won’t complain even if it’s wrong.
http://www.elephant.org.il/translate/translatable-but-debatable-patience-of-paper
So writing a D&D adventure is just like writing a fiction book, you are even called the MASTER and the rules say that what you say goes even if it defied the rules.
In a M&M adventure as opposed to this, it’s like you’re writing a non-fiction book in a hard scientific subject. You need to know what you’re talking about because the rules of nature cannot be changed.

But despite of what I just said, the Dungeon Masters in M&M can make “cross-pollination” / “brain-storming” about raw ideas, or help each other if they get stuck in the middle of an idea. Especially if they are in similar skill level in that particular subject.

Summary

Looking for what kind of games include math currently:
https://rpg.stackexchange.com/questions/28067/looking-for-a-game-system-that-involves-advanced-math
https://mathoverflow.net/questions/13638/which-popular-games-are-the-most-mathematical
You get to the conclusion that the game has two ways to do this:
• The game is based on a simple situation that has lots of possible combinations and in some way the rules are simple enough so that you can make an algorithm for a computer to play this too (against the human).
• The game is simulating the physical laws very accurately, or even if it’s different laws modelling them with precision so that you get the same result for the same initial conditions.

I’d like to believe that M&M can open a third category: not algorithmic and not simulation, but instead carefully hand crafted physical puzzles (in fact, any-natural-science puzzles).
I think the main requirement from the M&M Dungeon Master is to be good at
Analogical models
https://en.wikipedia.org/wiki/Analogical_models
like a very nice example is explaining electrical circuits as water pipes with the battery as the pump and a diode as a valve and so on.
https://en.wikipedia.org/wiki/Hydraulic_analogy

So I hope after reading this book someone will take up the gauntlet and make it into a proper game!

Nir Strulovitz, July 2021.

Chapter 0 – The Science of Sword and Sorcery

I’m watching now a documentary about the art that is inspired by D&D:
Eye of the Beholder: The Art of Dungeons & Dragons (2019)
https://www.eyeofthebeholdermovie.com/
So less than a minute into the movie, someone says:

There is suddenly a beholder in front of you.
“Well, what’s a beholder?”
“It’s a really angry onion with eyes and teeth and ‘tentacle-y’ stalks.”
And then you show them this very cool picture:

Again all rights reserved to Wizards of the Coast, and to the website of Amazon, and of course the talented artist Raymond Swanland https://raymondswanland.com/

So it made me read in Wikipedia to find out what this creature is all about:

Beholder (Dungeons & Dragons)
https://en.wikipedia.org/wiki/Beholder_(Dungeons_%26_Dragons)
According to the description this “ball” doesn’t bounce or roll: it floats in the air! But how?
It doesn’t seem to have wings or propellers, and it’s mouth is open so if it was like a big balloon of lighter-than-air gas:
Lifting gas
https://en.wikipedia.org/wiki/Lifting_gas
Or even more exotic vacuum:
https://en.wikipedia.org/wiki/Vacuum_airship

This raises (forgive the pun 😊 ) two questions:

  1. How come this light gas/vacuum doesn’t escape from the creature’s mouth? (and if it’s only concentrated in the back side of the creature, how come the back side doesn’t float higher and rotate the creature’s face down?)
  2. How does the creature move front back and to the sides? Also how does it turn around? If it can’t rotate at all then all the adventurers need to do is to run around the closest corner. A hovercraft uses rudders when it’s fast and differential side propellers when it’s slow (like if the right-side propeller pushes forward and the left-side propeller pushes backward (rotates in the opposite direction) then the whole body of the hovercraft will turn to the left.) Spaceships use small jets of compressed gas that they shoot sideways. If the eyestalks would double as elephant-like trunks this could be achieved, but they only have “eyes”.
  3. How does it avoid hitting the ceiling or flying too high? Possibly it can compress more or compress less (decompress) the “pockets” of light gas in it’s body.

OK. So we turn to Wikipedia again, this time to see if it can help us with how to levitate:

https://en.wikipedia.org/wiki/Levitation

OK so from all the forms of levitation: magnetic, electric, etc. I think the most likely to be at work here is buoyant levitation: this creature floats probably because it’s tissues contain pockets of some lighter then air gas like hydrogen.
Where does it get hydrogen gas? We know that cows’ farts make much of the methane that later rises to the atmosphere (methane is a green-house effect gas, about 30 times more than CO2 !), so I assume like our (and cows’) farts, the beholder’s body also produces methane, and maybe this creature found a way to break out the hydrogen inside the methane, perhaps using oxygen:

https://en.wikipedia.org/wiki/Hydrogen_production#Partial_oxidation

Then the creature stores the hydrogen in it’s upper part (the upper part of its ball shaped body, behind its big eye). So this takes care of not flipping over, because then it’s stable (his lower body contains no hydrogen, so it stays down).

About the method of navigating, I assume the creature has small nostrils (like in our nose) where it exhales the air that it inhales through its big mouth. The creature is able to direct each nostril up-down-and-sideways so each nostril act like the short gas bursts that a spaceship uses to turn around in space.

So for example the beholder of the type “Eye of Flame” , Wikipedia says that:
“These beholders serve more powerful beholders. They have fire rays, telekinesis rays and fear rays. When they die they explode in a fire burst.”
OK so now we understand why the explode, because the fire touches the hydrogen!

So suddenly we have things to calculate, like we need to calculate the volume of the creature (let’s say we know the upper quarter of this “ball” is filled with hydrogen) so we can calculate this from the creature’s diameter.
Why do we want this? Because on the one hand our heroes want to blow the creature to bits. This is done in two stages (like you need two gun shots to blow up a car) : one is to make the creature leak gas and the second to ignite the dispersed gas (hoping that now there’s 18.3% – 59% hydrogen and the rest air (with the needed oxygen).
https://en.wikipedia.org/wiki/Hydrogen_safety#Mixtures
“The limits of detonability of hydrogen in air are 18.3 to 59 percent by volume”

But on the other hand if we’re in a closed space (like a dungeon) the blast wave will kill us too.
https://en.wikipedia.org/wiki/Blast_wave
There are two kinds of explosions:
Professional explosives like TNT and C4 are called high-order explosives. They are more powerful, they detonate to produce supersonic blast wave.
Improvised explosives like gunpowder and petroleum are called low-order explosives. They are less powerful, they deflagrate to create a subsonic explosion without blast wave.

So as you understand our creature’s explosion belongs to the low order explosives.
Another thing you need to know is that it makes a lot of difference whether you’re outside or inside. When I was a teenager I used to train in a group activity of kung fu, and my master (Udi Ra’anan) was once in the marine commando unit (Shayetet 13), so one day he told us by the way (hypothetically) that if a grenade (TNT) would explode in the training hall then if we lay on the floor we’ll survive, because there was an open door, so it’s not an enclosed space. But if someone would have closed that door beforehand, even using a thin sheet of nylon, then everyone will die from the same explosion.

But for simplicity let’s say that we’re shooting the creature from a safety distance using flaming arrows, how far should we be to survive (distance in meters; weight in kilograms)?

distance=〖130*(weight)〗^(1/3)

Explosion Danger Area Calculator
https://unsaferguard.org/un-saferguard/explosion-danger-area

OK let’s take another example of trying to think HOW IT WORKS.

D&Diesel with Vin Diesel (Extended Version)
By
Geek & Sundry
https://www.youtube.com/watch?v=yLEMb_RIZ3o

How can Kaulder the Witch Hunter (race Human) be an 800 years old immortal?

OK, so for this I watched two documentaries:
The future of work and death (2016)
Immortal A Horizon Guide to Ageing (2012)

It turns out there are only two animals in nature that can live forever (if they don’t get injured or sick). Both of these animals happen to be from the phylum (“family”) Cnidaria, which are very simple animals that live in water, they have no eyes or other body parts, that sting their pray. Both are very small, their length varies from half a centimeter to one centimeter.

The first animal lives in the sea and it’s a jellyfish called Turritopsis dohrnii a.k.a. immortal jellyfish, it is able to revert back to its young stage (polyp), so this means it can live forever!
https://en.wikipedia.org/wiki/Turritopsis_dohrnii
The second animal lives in fresh water and is called Hydra. I guess you know the D&D monster that was copied from the Hydra of Lerna in the Greek mythology – The mythological Hydra possessed many snake-like heads, which all have a regeneration feature: for every head chopped off, the Hydra would regrow two heads. Isn’t it much more awesome that there is actually a creature like this in nature?
https://en.wikipedia.org/wiki/Hydra_(genus)
The real life hydra can grow a whole body even if you cut it into 20 pieces! How cool is that?

There’s a much bigger animal – Salamander (an amphibian creature which looks like a lizard) – which also have this power in a more limited way: it can do limb regeneration, like growing back a complete leg! But a Salamander only lives for up to 50 years.

So when trying to read how this works, that specific type of jellyfish turns itself to “before puberty” stage, While the hydra just doesn’t get old:
Hydra stem cells have a capacity for indefinite self-renewal. The transcription factor “forkhead box O” (FoxO) has been identified as a critical driver of the continuous self-renewal of Hydra.

So what is a “transcription factor”?
It’s like in a cookbook when you have a recipe that is divided into two parts: how to cook that thing in the oven, and then how to cook that same thing in the microwave. So of course, you shouldn’t do both, you only perform one of these methods, depends on which appliance you got in your kitchen. So you read the whole part that’s relevant to you (for example: oven method), and you skip the whole part that’s not relevant to you (for example: microwave method).
If you know how to program for example in C or C++ language, it’s exactly like the macro preprocessor: all that stuff between #ifdef and #endif.
So a “transcription factor” tells us: do this part of the code, don’t do that part of the code. But instead of a recipe algorithm or programming language code, it does the same for the DNA code.

If you want to see visually how transcription factor work, you can watch this very cool video by Blair Lyons from Stroma Studios, She also turns molecular biology into 3D video games (see in other videos by her):
How Genes are Regulated: Transcription Factors
Youtube version:
https://www.youtube.com/watch?v=MkUgkDLp2iE
Vimeo version:
https://vimeo.com/30034882

Is it possible that thig guy Kaulder the Witch Hunter has genes from this jellyfish?
In nature different species cannot procreate, that is the definition of a different species. So even if somehow a woman would get the sperm of such a jellyfish, this sperm cannot fertilize her eggs. Plus we know that Kaulder the Witch Hunter doesn’t look half man half jellyfish.

OK. Is it possible that Kaulder the Witch Hunter uses only a few of the jellyfish genes, in an artificial way? Today scientists can do things like that – although it’s very dangerous and morally wrong (for example, think what would happen if evil dictators lived forever), using a technology called CRISPR which is like copy-paste for genes. So maybe an evil witch made experiments on Kaulder’s mom and changed one of her eggs before Kaulder was born? This can also explain his hostility towards witches!

Let’s try to calculate what are the chances that this exact gene change will happen just by chance (mutation).

First of all why do we think it’s just a single gene and not several genes? Because Wikipedia says that scientists already found this specific gene:

https://en.wikipedia.org/wiki/Turritopsis_dohrnii#Genomic_analysis
Genomic analyses such as sequence analysis on mRNA or mitochondria DNA have been employed to investigate its lifecycle. mRNA analysis of each life stage showed that a stage-specific gene in the medusae stage is expressed tenfold more than in other stages. This gene is relative to a Wnt signal that can induce a regeneration process upon injury.

https://en.wikipedia.org/wiki/Human_genome#Information_content
The haploid human genome (23 chromosomes) is about 3 billion base pairs long and contains around 30,000 genes.

On the other hand we have the mutation rate:
https://en.wikipedia.org/wiki/Mutation_rate
The human germline mutation rate is approximately 0.5×10^(-9) per basepair per year.

So now we will only look at the base pairs, which are the letters of the genetic alphabet. The DNA doesn’t have 26 letters like English. The DNA has only 4 letters: A, C, G, T.

So let’s take the simplest scenario, let’s assume that a specific change in just 1 of these 3,000,000,000 letters is enough to make Kaulder the Witch Hunter immortal. What is the probability for this to happen? So in words our formula will be:

immortal probability=
probability to pick right letter
×probability for change in letter
×probability for right change

When we have probability (chance) of “this needs to happen” and “that needs to happen also” and so on, we multiply them.
The probability (chance) of picking the right letter is one out of three billions or in numbers we write one over three billion like this: 1/3000000000

The probability of a change (mutation) to happen in any given single letter during a whole year, is 0.5×10^(-9) or if we want to write in common fractions (simple fractions) it’s 1/2×1/1000000000
(when we have a MINUS in the exponent, it means that it’s ONE OVER that same thing but without the minus).
The probability to pick up the right letter from A, C, G, T: 1/4

OK. So finally let’s put everything we have into our formula:
immortal probability=
1/3000000000×1/2×1/1000000000×1/4

The result is a TINY number: one over a HUGE number! (we could guestimate by adding the zeroes)

1/24000000000000000000

This means each year ONE out of such a HUGE number of people would have the exact mutation in the exact place to become immortal.

But, I hear you say, we are not talking about just one person, and not just one year! Kaulder the Witch Hunter is human, and the human race is made of lots of humans over many years.

OK. Since we don’t have exact information about their imaginary world, let’s assume it’s like our world. Humanity started with some biological “Adam and Eve” and they lived 200000 years ago. At the beginning there were very few humans, but let’s “meet them half way”, today there are almost eight billion people, so let’s work with the number four billion: 4000000000 people.

So we multiply these two sizable numbers in the numerator, to make the fraction less tiny:

800000000000000/24000000000000000000

So after simplifying (reducing) this fraction we get:
1/30000
Which is the probability for any one WORLD with humans to develop one IMMORTAL human!
So we have an empire of thirty thousand planets, one of them will create immortal humans*.

*But before you get too excited, try to think about what happens if the immortal property is determined by not 1 letter but instead 2 letters. This means that we multiply by 1 over 3 billion not once but twice! That’s 18 ZEROES in the denominator which is a lot.
Now think that if it’s a whole GENE that controls this property, 1 human GENE is made up of 600 letters! So then we need to multiply by 1 over 3 billion not once or twice, but 600 times – this means roughly 600×9=5400 ZEROES in the denominator. Remember when we called 18 ZEROES a lot? So what do we call this? So the more realistic scenario with a whole GENE explains why this probability (that humans will develop immortality from natural mutations) is practically ZERO.

OK. But you see that even very basic concepts in D&D lead us to interesting calculations!

OK let’s see another example.
All rights reserved to Wizards of the Coast and to Jeff Easley the gifted painter.
https://jeffeasleyart.com/

This image was the cover of a booklet by TSR that explained the magic in Forgotten Realms.
https://en.wikipedia.org/wiki/The_Magister

So we see the wizard holding up his hand, and it looks like a ball of lightning in his hand. The picture is not explained inside the booklet, maybe it is the Fire Gyregam, which is a metal ball that spins fast and shoots lightning bolts. So for simplicity let’s talk about the lightning bolt spell. https://roll20.net/compendium/dnd5e/Lightning%20Bolt#content
This is supposed to be along a line that you can aim in the direction of your target. So the attack is along a cylinder path which is 30 meters long and 1.5 meters in width diameter.
Although there is a way to force an electric arc to go in a specific direction, by firing laser beams at the gas between the electrodes, this doesn’t seem to happen here.
https://en.wikipedia.org/wiki/Electric_arc#Guiding_the_arc
another way to create a long arc (60 meters) is using a very thin copper wire (a taser gun also uses copper wire in a different way) that will explode when the current pass through it. But again it doesn’t look like that’s what’s happening here.
https://phys.org/news/2011-11-extra-long-electrical-arcs-energy.html
If we simply create a very high voltage (like with a Tesla coil) then it we cannot tell it in which direction to go: the current will flow through the body of the wizard himself into the ground, and he will be killed.
He can prevent this if he is completely covered in chain mail armor that will act like Faraday Cage like in this nice demonstration:
When Tesla Met Faraday
Association of Science and Technology Centers
https://www.youtube.com/watch?v=WFl2yiw9nek

Anyway let’s try to calculate how high is the voltage that we need to use if we cannot control the direction of the electrical breakdown of the air (that’s when the air becomes conductive by becoming plasma because of very high voltage). We calculate this using Paschen’s law
https://en.wikipedia.org/wiki/Paschen%27s_law

V_B=(Bpd)/ln⁡〖(Apd)-ln⁡[ln⁡(1+1/γ_se ) ] 〗

To be honest I found the explanation in this website much clearer:
Sciencing: where science is relevant and fun for everyone!
Calculating the Breakdown Voltage of Air Per Inch
https://sciencing.com/calculate-breakdown-voltage-6728339.html

by the way if you want to get intuitive understanding of how it works physically, then check out this nice video:
Plasma and Paschen’s Curve
David Shlivko
https://www.youtube.com/watch?v=r66nrK9R4Z4

A=112.50 〖(kPa*cm)〗^(-1)

B=2737.50 V⁄((kPa*cm) )

γ_se=0.01

p=101325 Pa

d=30 m