Speedy is a Nerd

This Thread is just for me to post things I overthink about in general.  And yes, I am a pretty big nerd I guess.


I really love chess and recently I made custom theme for chess.com pieces that are all Riverclan and Skyclan due to their heightened tensions:

In addition, I wrote up why each cat got the position on the board that I gave them:




Art Credits:
Xaan drew/created all the riverclan cats <3
Maple drew/created the skyclan pawns

Stormwatcher was drew/created by copperbeans
Sunfire was taken from J's signature
Beech was drew/created from xaan
Wolf was drew/creataed from kiikko
Bird was drew/created from xaan


(The riverclan pawns were the emoji from them on discord and I do not know their author however?? For some reason I thought it was maple.  This is not correct I later learned, if anyone knows their author I can update the credits for that For now I will just credit that under the discord.)


The ROOK
Storm is the rook because he is sturdy and powerful.
Storm often provides back up to his younger siblings Rowan and spruce.  The rook usually guards the bishop and knight from a distance in chess.
Additionally, the rooks and queen can team up to create batteries in chess where they all target the same piece.  This is like how since storm is a senior warrior he would work well with the deputy queen poppy.

The knight
Rowan is the knight because he is crafty, witty and a bit sneaky at times. The knight is known for sneaking around enemy positions and can fork enemy pieces leading to victory. Early in life him and Sprucebark often butted heads, in chess this is represented by the knights and bishops often balancing one another and the argument for which piece is better. Now that they are close friends they make a powerful team.

The bishop
Sprucebark is the bishop because he is so very rule oriented.  Everyone knows spruce has certain quirks about him and sometimes his interpretation of the warrior code makes him see things as black and white, in chess the bishops always stay on either the black or white squares so this piece really does fit him perfectly.

And here is Another nice alignment here with him and his brother Rowan; in the beginning of chess the bishops start closer to the king and queen than the knights, this represents how spruce usually tends to stay closer to poppy and perch and Rowan is (more so) an exploring spirit, sometimes going off on his own more.

The Queen <3
Poppyshine is the queen because of the power it has.  She is the life of the clan and her protecting all the other pieces seems very realistic.  An additional note:  the queen and knight are sometimes regarded as a very good pair of attacking pieces, this represents the close bond poppy and Rowan have

The king
Perchstar is the king because well because she is leader obviously.  But ok going into a super overly complicated reading wayyyy too much into it there is more too…

In chess the king is weak in the beginning of the game.  But as the game goes on for more and more time, the king gets stronger and stronger (assuming it’s a balanced ish game that makes it to an endgame) this represents that early in perch’s character arc she often doubted herself.  But with time and the help of poppy she has become much more strong as the “game” has
Progressed.

The pawns:
I chose the riverclan symbol here because I couldn’t think of anything else that would represent 8 different items.  But since this is a deeeppp dive I feel I must give some explanation to it:  the pawns in chess individually are very weak but in pairs or trios they can become quite strong.  This doesn’t represent any one character but rather the ideology the clans have of staying together strong and having one another’s back.  In chess when a pawn reaches the end of the board it can become a new queen. This is somewhat like them getting a warrior name but okay not exactly since the pawns represent more of the clan ideas and so on but it is a close try. So they can be thought of as the clans ideology OR as apprentices that will one day promote to new pieces once they reach the opposing side of the board.  The pawns are the same idea for both clans so they will not be repeated under skyclans section




The ROOK!
Stormwatcher is the rook because in chess, the rook is an endgame piece.  It only comes into play after the initial battle is under way because it starts off in the corner.  This is very representative of Stormwatchers life.  His early warrior days were spent with him lazing about and doing as little to help as possible.  Once Wolf (The queen) sorta got on his case about this he finally becomes a more active warrior in the clan.  Just like the rook he is not involved super early on but is vital to winning endgames by gaurding the pawns as they push towards becoming new queens.  This is also representative of how Stormwatcher is (usually) very protective of apprentices : Such as rescuing lightningpaw from the twoleg cage or how he secretly is actually very proud of his apprentice Stone


The Knight:
Sunfire seems like a very good knight because he can get into troubles (see how he started the rift with thunderclan).  The knight is an early attacking piece and is actually the ONLY piece to be able to move on the first turn (other than pawns).  The initiative the knight has to be able to do this matches sunfire's energy to go take charge even if that means going to go hunt on land that is not his.

The Bishop:
Beechfang is the bishop because that is what this whole war is about.

(perhaps I could have also made beech the king since if beech dies it is like the black pieces being checkmated since that is skyclans objective of this coming war, to protect beech)

But nevertheless, I made her the bishop because she is the direct piece counter to sprucebark.  Sprucebark and her have directly opposing views on the world, she is okay with 'coloring outside the lines' so to speak which is mirroring opposite to Mr. Rules.  In chess, the bishops on the same color complex are often traded off as they fight for the best diagonal lines.  

The Queen! <3
Keeping in theme the deputies just seem to be good queens overall.  Wolf similarly is the lead warrior for her clan so it does make sense for her to be the queen here leading the rest of the pieces in defense of skyclan and beech in general.  

The king
Birdstar is the king because she is dictating that the war is going to happen.  Like it or not fall in line everyone! She is the one who hid beech here and choose to take the risk of this being discovered.

Objectively speaking should the leaders have been the queens due to them being the most powerful?? Perhaps, but you go make the pieces if you wanna then (joking).  But still, the king in chess is vital to the endgame because sometimes it is the last major piece you have left and the leaders are unusually the ones who will not die in a war because of their multiple lives... well until they are checkmated.




Additionally:
It was said in discord by down that bird would not be making the first move aggressively in this war.  In chess the white stones move first, so this is correct given that we will likely see riverclan making the first play in this.


To say that I am excited for this showdown is an understatement.  I can not wait to see some pieces go flying and hopefully we get some cool sacrifices.  Perhaps I will even turn the battle we get out of this into a chess game.  That would be pretty nerdy indeed but I could totally seeing myself doing that and annotating the whole thing with detailed explanations and breakdowns of the moves as well as computer evaluation of why a certain move is terrible or genius XD

-I will probably post more nerd stuff like this in this thread over time

you never know what you will get in this thread, will the next post be about chess or about a deep analysis of something else equally as silly?  Who knows

If anyone wants the code that does this, feel free to msg me. I am not gonna include it here because it is super long.

Today's random thought experiment:

(and yes... I overthink things to the extreme don't argue, you are the one who clicked here!)


Are there more possible chess configurations VS are there more minecraft inventory loadouts


Hmm an interesting question. But how do we solve it? Well I have no idea, the question at first seems very daunting but we can try and break it apart and I will do my best to do just this, whenever you get some random mathy question like this I try and break it apart into its simplest terms and work from there... so lets start with what we know:

There are, 6 different chess pieces on each side for a total of 12 if we think about it

So is the answer as simple as 64 choose 12? NO! it is not because this would lead to positions where black and white could both have multiple kings so we cannot allow for this, it could also lead to positions where both kings are in check at the same time which we also find as an illegal position.

So how do we calculate this?

I have 0 idea of how to get the actual exact answer and even so the answer would be so mind numbing large that it will not even mean much to look at. It would have to be put into scienctific notation and even then it would not even make sense...

So what if instead we think about the problem as slots in our minecraft inventory vs the chess squares on the board.

The 27 storage slots that the minecraft game has would at first seem to put it at a disadvantage... HOWEVER!! minecraft has so so so many items and each (or rather most) can stack up to 64 so that means that if a player was holding one carrot, that is an option but if they hold 2 carrots now that is a new item in our set. A set is defined by UNIQUE things, not their order. So if a player has 2 carrots but another player comes along also only having 2 carrots just in a different slot we are counting these players as equivelent, but counting them as different would be even more ludicrous and interesting.

So really... is it 27 * 64 = 1728 choose the number of minecraft items? I think NOT, because it gets tricky because some items only stack to 16 and then again a bow does not stack at all.

And what about enchanted items? A flame bow 1 is different from a bow and that is different from a flame bow 2...

PLUSS... think about THIS!!! there are 2,700 possible tropical fish in the game and these can all be put into a bucket that the player could hold meaning we need to multiply whatever answer we currently got by that.... but the player could have more than one tropical fish bucket in their hand... in fact they could have 27...


AND HERE IS WHERE WE REALLY FALL OFF A CLIFF...

SHULKER BOXES!!!!!

A shulker box has a single shulker box adds an additional 27 slots of space that the player could hold and if you cram your inventory full of them and they can all be different colours too!

meaning we now have... ummmmm

729 base slots to work with but since a yellow shulker box is different from a green one we need to add the colors of shulker boxes to what we have...hmmm

there are 16 dye colours a shulker box can be

hnggggg my goodness! So a matching inventory that would all else be equal could be set as a new item in our set by just changing one shulker boxes colour and we can do that across all our 27 base slots.

It should be clear that the number of possible minecraft invetory loadouts VASTLY trumps the number of possible chess positions.


Some reasearches have already proven that there are more chess positions than there are atoms in the universe so it follows that there are VASTLY many more minecraft loadout inventory options than atoms in the universe...

So next time you load up the silly little block game remember that you are carying one out of a (soft infinite) number of possibilities.

Now... why do I say soft infinite.. well because even though I cannot calculate the number as you see it gets mind numbingly complicated very quickly it is still some number.... there is a limit to where you will start to repeat loadouts thus meaning there is some number n such that n is some gross long number that we might not even have a name for but it is out there... somewhere way down the number line.


Maybe you are wondering, why speedy? Thats a good question, idk... for funsies?

Its also interesting to me that there are some infinities greater than other infinites, that boggles my mind but when I see people prove it, makes sense and it checks out from everything we know about math so yeah there you have it I think...

I THINK... there is more minecraft options than chess positions and I think they are both still numbers though discustingly large to the point we wouldnt even be able to really compare it to much else.

And even still not infinate, and infinity is not even the biggest thing anymore either so hmm

what will you use this for? ... probably absolutely nothing at all! but it was fun to write! maybe byrin will read it and shake her head disapprovingly and tell me I am wrong in some aspects of this (she is math prof) and that would be very welcome!

OHHHH ok... so I thought about this more while on a short walk outside in the woods... (it was cold out and now my toes feel all numb)


But I think I got it!!!!!!


Alright lets first give a more formal proof that the minecraft inventory is greater than the chess positions, we can actually do this quite easily and im kicking myself for no thinking of it earlier...


Let us, map each chess square to a slot in 3 shulker boxes.

We need 3 shulker boxes because 3*27 = 81
2 * 27 = 54 and there are 64 chess squares.


So we have these 3 shulker boxes where
the first slot in the first shulker box represents the chess square grid A1,
the second slot in the first shulker box represents A2 .... so on and so forth.


Now all we need to do to represent all chess positions is assign each chess piece to a minecraft item.

White king = fishing pole
Black king = white concrete... and so on and so forth.

We can now replicate ANY chess position and we can do it with just 3 shulker boxes.

This is a much more valid proof than what I had earlier which was much more of a guesswork and gut feeling type of approach.


Since we could also have 4 shulker boxes in our inventory it is very quickly seeable how it is correct that the minecraft slots have way many more possiblities



HOWEVER, can we do any better on how many this is??

Well... sort of I can give an upper limit to it like this:


1) whatever the maximium size of the player inventory file is, (for arguments sake lets call it F for file size)

take F and figure out how many characters this is, lets call this C for characters,


All it is now is

C choose the number of valid characters this file type can have!

BOOM! Now we have a hard upper limit meaning that no matter what minecraft inventory you throw at me I will be able to describe it with that amount of information available to me to write to the file F.


HOWEVER... why is it an upper limit speedy??

Well, because it is describing things that are not valid minecraft inventories as well....


For instance thinking of our file:


file option 1 might look like:

{one carrot in slot number 2}

(obviously the encoding that mojang uses is different but this gets the idea across)

so this file option 1 is a perfectly valid thing, minecraft will load up and know what to do here


However, our upper limit allows for things like

file option 2 here:

{hello world, speedy is a nerd}

This is also a valid file that we have described however now minecraft will load up and wonder what in the world a speedy is and probably crash or thow an error,


So the upper limit

C choose valid characters on a keyboard gives us an upper limit that can map to all minecraft invetories however it gives things that do not map to the minecraft valid inventories as well meaning it is stricktly bigger set.



So can you do any better speedy??

Well... hmm maybe?

What if we think about other common problems people do with these types of things... I was thinking about a deck of cards.

51! is very easy to see this is the number of ways a deck of cards could be arranged.

all that is is a fancy way to write (51 * 50 * 49 .... ext)

why is it this? Well think about it, the first card slot has 51 valid options and so we already split our tree branch off 51 times however now we have locked a card in meaning each of these 51 branches can now only split 50 times, and we continue till 1)


so... minecraft can easily map all card deck possibilites too with a similar argument as above for chess, just give to me 3 shulker boxes that can represent each of the 51 card slots and I can assign some minecraft items to each of the card values. EZ peasy!


Now... for the number of minecraft inventory slots... yikesssies.


Well... lets us think about it like this.

Take the number of minecraft items. Let us call this I

BUT, here is an important distinction, its not the number of items but the number of possible items you can have in a slot! Meaning I will be the number such that 1 carrot is different from 2 carrots, this way we can erase the problem of having to multiply by 64 or16 later on depending on how big an items max slot is.


So we take this I and I can be put into 27 base slots so it becomes

27 choose I

But what about those darn shulker boxes?

Well, now it becomes...


729 choose I... but this number only describes the possibility of us using all shulker boxes possible (we could have a base slot not being a shulker box yes?)



So it really should be put into a loop and math has a loop built into it for us called summation.

So I think it should be..


Summation of ( 27 * n choose I) from n = 1 to n = 27

Because now we cover the cases where we do not jam our inventories full of shulker boxes as well.

(In english we are looping over each possible shulker box we could add and multiply that by 27 which is the shulker box slots then choose I out of that where I is described above then just add all of those together)

I really think this gets you close... HOWEVER... it does not account or describe if I should include a shulker box or not in its count

We need to get the shulker boxes in the base slots but a shulker box cannot contain another shulker box so we can do instead...

Summation of ( 27 * n choose I) from n = 2 to n = 27

+
Summation of ( 27 * n choose J) from n = 1 to n = 2

Where J is the set I plus the 16 shulker box variations this way only the base level slots are able to include shulker boxes in them.


So there is your answer... I think?