Bingo! I had originally planned to show off this checkerboarding pattern, but for the sake of visuals I split it up the way I did. This means technically that the chain in the video is actually *not* ergodic but rather 2-periodic. Splitting it up 50/50 does make it settle on a single steady state though (and looks better on video)!

I like your funny words, magic men 😅

what words did i just read

Having learned just enough linear algebra to only half-understand this gives probably a worse feeling than not understanding it at all

@@Chubby_Bub Go to the numeric simulation and pause. There are a bunch of numbers on screen that represent the probability a ghost is at that position. Shove em all in a vector. In this case it's like a 500-dimensional vector but you can visualize it in 3D as long as you don't try to do something insane like take a cross product. So you've got a vector for the state. Better yet, you've got a space that represents every possible probability-state. (Exercise: What is the natural basis for this space? Is it orthonormal? What do those basis vectors represent?) Now, you'll have to hold both the visualization and the simulated interpretation in mind at the same time for this next bit. Take a single simulation step. _Click…_ The vector changes. _Click… Click… Click…_ Every step makes the vector jump a little bit. (Exercise: What property does every state vector have?) In fact the step function is a linear transformation. You can write it as a square matrix. If the simulation converges, that means that repeatedly applying the linear transformation to the initial vector converges to some other vector. And if you find a vector which is invariant under the transformation, you've found a steady state of probabilities. That's the definition of an eigenvector. By representing the state transition as a matrix you can just do linear algebra to figure out its steady state, if it has one. But wait - there's more. Ghost walks in the maze have period 2: If a ghost starts on a tile and continues walking, at every point in time only half of the board is accessible. The probability of half of the tiles is zero. Those two smart people above are saying you can see this from the Jordan canonical form but I can't picture that because I took my linear algebra class like 150 years ago. Anyway, linear algebra is easy. All you have to do is imagine 500 dimensions and you're good to go.

when i saw that the output of the rng was literally determined by the first bit of the game rom it put a huge smile on my face, its so crazy what early programmers managed to accomplish with so little room to spare

​@@lavenderinthedark But they just made it worse by turning a perfectly evenly distributed RNG into a heavily biased RNG. EDIT: Actually the original RNG was bad too so indexing into the rom might be better...

​@@torgematthies2172 why was it bad? Too predictable?

Excellent Dave the Barbarian reference, btw.

​@@Vextrove Because it's sequential. It grows until it hits 8192, then drops and grows again. Indexing into ROM is an attempt to make it non-sequential. And is... mostly-successful.

Sounds just like all my rng research for speedrunners! 😂 It's really interesting though!! Haha

It's useful if you are trying to get the high score. Knowing which way ghosts are most likely to go at intersections while frightened makes it easier to eat them. Up to 12,000 bonus points can be gained per level from eating frightened ghosts. To put this in perspective, only 2600 points can be gained from eating all of the pellets in a level.

@@jimmyhirr5773 if you're going for maximum score, only 17 out of the 256 levels actually let you eat the ghosts. It's still a little useful to know they're more likely to turn left, but it only matters right at the beginning.

Actually, this is much more common than you might think. Consider as well that adding a table of sufficient size for this would add a significant total to the game's final size, further increasing it's production cost.

Uniformity is not the only desirable feature of a rng. n -> n + 1 achieves this but isn’t very random. n - 5n + 1 isn’t much better; since we only use the last 2 bits all the modulo junk disappears, and the rng literally becomes 01 -> 10 -> 11 -> 00 -> 01, which, while uniform, is very predictable. Even more since we have 4 ghosts, each pulling exactly one random number per turn.

tbh it is a hilarious idea

That's where Clyde likes to hang out anyway.

​@@klafbang Predictability doesnt matter because the position of the ghosts when they get scared isn't going to be consistent in the first place. Regardless, pulling a byte from the game's memory doesn't really solve this issue in the slightest, it doesn't make the function more random it just adds an extra step.

Easily his best video imo.

Maybe Pac-Man ghosts are the reanimated souls of the RCT guests that "mysteriously" drowned in the park's pond.

@@RGMechEx BWAHAHAHAHA!! *beautiful*

Marcel Vos is one of my favorite channels

You beat me to it. Oh, well. Having another comment that says roughly the same thing (albeit wordierly) on the same video won't hurt anyone.

Me too

The Atari 2600 game "Yar's Revenge" did the same thing. It's a fun trick. I'm sure yet others did this, too.

@@nstbayless It could actually work a whole lot better if they happened to have any compressed data in ROM, since it's highly entropic one could expect the distribution to be a bit more balanced ; I now wonder if Pokémon used the sprite data for the RNG

​@@cheaterman49 og pokemon was written by people with a passing understanding of the concept of programming, the way they stored and retrieved some information is responsible for the famous missingno glitch. I forget the specifics but it's wild, so many things in those games just don't work as intended because of the way they're programmed

@@thesquirrelisking I tend to cut them some slack, they were working on a system where doing things properly would have been too slow, they were literally chasing cycles haha

​@@thesquirrelisking hey, no offense, but if you think those Gameboy games were coded by people who had very little understanding of programming, I don't think you could hello world yourself out of BASIC. Anyone who's coded in html even once could tell you the amount of work that goes into a fullscale game like that, especially on proprietary hardware.

As someone who occasionally designs PCBs, I felt that. :-)

i can only imagine they wrote an algorithm to calculate all of the vectors for them. right? right??

@@durdleduc8520 I wouldn't be so sure. RGME has calculated Pokemon sprite decompression and drawn it out by hand before.

you can see this just from the function used, a_(n+1) = 5 a_n + 1. Looking at the bottom 2 bits here is equivalent to looking at each term mod 4. a_(n+1) mod 4 ~= (5 a_n + 1) mod 4 ~= (a_n + 1 + 4 a_n) mod 4 ~= (a_n + 1) mod 4. I'm using ~= to indicate congruence here, can't remember if I'm supposed to use the 3 line equals or not. This result indicates that when looking at the bottom 2 bits, the recurrence actually just always adds 1, never anything more interesting.

Now I actually kind of want to see how using this kind of RNG would impact the ghosts' movement.

since in many systems small changes explode exponentially, and (as far as we know) quantum mechanics isnt deterministic, this actually isn't true! for a coin flip quantum uncertainty probably isnt relevant, but for say, what the weather will be like in exactly one year? *literally* random

@@terdragontra8900 Interesting! I want to learn more about that, outside of the video games it's definitely not a concept Im familiar with

@@terdragontra8900 As far as I understand it, everything is deterministic, it's just often so astronomically complex there is no way a human could ever grasp it. i.e. randomness does not actually exist, it's just our way of describing events where the cause-and-effect chain is unclear to us. Quantum mechanics includes uncertainties for states but this is just a mathematical structure to describe the behaviour of systems, since there is no way to directly observe interactions. The actual interactions play out in a deterministic fashion.

@@bluegum6438 Read up on "hidden variables" and have your mind blown. We've proven experimentally that entangled particles *do not* decide how they will be measured in the future when they are created.

For a more concrete example of true randomness in nature, IIRC it's been observed that it's completely random how far an electron will be from the nucleus of its atom when measured (though the distribution isn't uniform, they tend to hover around predictable distances, but they're still not guaranteed whether or not they'll appear at that distance at all). Another example is how it's random when a radioactive particle will decay, which means that theoretically a lump of radioactive material could at any point just decide not to be radioactive for a full minute or so.

1 much, much worse. You’d have a function with period 4, from which you’d pick 4 random numbers per cycle. Each ghost would be 100% predictable.

@@klafbang Are you sure? I'm not referring to just 1,2,3,4,5... but the 1,6,31... (the index as explained in 3:26). The current rng is also period 4. I guess it would be more similar (if not the same) as the 25% test

@@TrianguloY the current rng is period 8192; the modulo matters when looking up in the rom. It might be 00 -> 01 -> 00 -> 11 (or whatever). The index you’re using as input is not directly linked to what you output, sou can have a longer period.

It works out like that because of modular artithmetics. Basically, you can move the mods around wherever. ((5n + 1) mod 8192) mod 4) = (5n + 1) mod 4 (as 4 divides 8192) = (5(n mod 4) + 1) mod 4, so the computation without going via rom becomes a simple operation on only the two bits of output, giving it at most 4 possible values.

Oh! I see it now. Since 8192 is divisible by 4 the output is 5n+1 (mod 4) and with modulo operations you can simplify it to (5 mod 4)n + (1 mod 4) which is n+1! So the sequence is 0,1,2,3,0,1,2,3... *alternatively 4n + n + 1 (mod 4) so 0+n+1

They spend several seconds frightened in the first level, and then the length of time gets shorter and shorter each level.

The exact same behavior occurs when eating a power pellet, just the length of time they are frightened gets shorter and shorter until they flash, turn around and instantly are not frightened.

In Ms. Pac-Man, the duration gets shorter and shorter by level until it's only a frame or something, but then it gets longer again and then it gets shorter faster. At least, with the factory settings of the tables I've played.

I would have to agree!

Pac-Man is quite representative of how "AI" usually works in games: either a deterministic set of instructions, or pseudo-randomness, or some combination of the two.

I imagine how it works (at least for Galaga) is that while the ships are flying in a line, they're treated as one entity for movement purposes, kind of like the swarms of bats in Breath of the Wild. At least, that's what I'd do. Couldn't tell you for a fact.

It made it better. If they just used the index, the ghost movement would have period 4, and the ghosts would just move around in a really obvious pattern. But since they used the entire index (as opposed to a couple bits of it), the ghost movement has period 8192, which is a lot less obvious.

One could argue they are frightened for exactly zero frames.

Balancing the Rom numbers would just push the probabilities of each direction closer to 25% so all in all, very little would change.

So for all practical purposes, dice and coins might as well be perfectly random.

Except the fact that atmospheric noise is random makes dice random as the forces from the atmospheric noise affect how the dice moves.

It's all mostly After Effects!