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Yes! You have it. Exactly: if you have no further information ("no theory" in your words) you can ask 2 questions, hoping that you will find the number quicker, but if you are unlucky, there is a higher chance that you will need to ask 4 questions. So 3 is the overall optimal: it's always 3, no risk. There is no way to solve it with 2 question without risking. If you have some information ("theory"), that usually it's AB, and rarely it's CDEFGH, you can go for AB first. But in this case the symbols are not equiprobable so it's not random. And we are going for random content, so you don't have more information what the result usually is (what distribution the symbols usually follow). If we do the game with 16 letters, it's the same: the optimal is 4 questions. With 32 it's 5. No matter how big your range is, the optimal way is always log2(n). Going even higher doesn't change the game: If you have 4096x8 random bits, your optimal way to find out those bits is to do it in 4096x8 questions. Can't do it in less. Practical compression example: try compressing random files with any compression software. The result will always be a bit bigger. In case of paq* it will actually try finding patters (find a "theory" behind the bits and bytes). It really does try finding patterns! Being a random file we know that there are no patterns there. But paq does not know that. So it tries. When it believes it found some theory (that some symbols are more probably than others) then it applies it - and experiences a loss. Ouch! So the result gets a bit bigger. Like trying to guess the 8-symbol game in 2 question. Risky right? Because it is programmed to lower the risk and try finding the best theory it will give up soon trying to solve the 8-symbol game in 2 questions - it will stick with 3 questions. And so it reduces it's loss. So this is the reason why there is always a small loss there - because it is always doing a bit riskier at the beginning, so those losses are inevitable (in case of paq or any other compression software with similar internal workings).

xinix you are a bit harsh. As Trench said: "upset". I know it's upsetting when people have claims or statements and they don't fully understand the subject (yet), but it's OK. I sense that he's open and he tries to understand it deeper, so let's give it a go. He has no background in programming or information theory - it would be nice to have, but what can you do? We are here to help him (remember: he came with a question). This is what we can do: practical examples, visualizations, games - so even without the full experience or solid background at least he can sense what randomness means. You know how I started in ~ 2007? Funny: I tried to see if I could compress the already compressed paq8hp result smaller. I could easy-peasy win the contest. Fortunately I have math and programming background, and I quickly realized the it won't work. You do have to try compressing random files in order to understand what randomness means (not with paq or any other software - they don't give you first-hand experience, but in your own way, creating your own tools). So then I deeply understood the real meaning of randomness. And I can tell: it has its beauty. Whatever you do, however you slice, you just can't make them smaller. The result will always have the same size (in optimal case). So I tried it, and have actual first-hand experience. Equipped with my math background I also know the explanation. Now he can have a tiny-little experience by trying to solve the ABCDEFGH-puzzle in 2 questions.

I'm trying to understand what's happening. So, we had breakthroughs in 2007, 2015, 2020, and now in 2021. The messages switching between significant savings and "little smaller". At this point this is what I get: no breakthroughs, the previous attempts were false alarms. This time you can probably save a couple of bytes with not completely random files. (Which is belivable). Your request for "be patient" tells me that you are more careful this time. I think it's wise, and I respect that.

Ahaaaa!!! There you go! Kudos to RANDOM, you had an epiphany! We wrote you too that your "random" 4kb file is also very small for correct tests. You need at least 1 megabyte! That would eliminate some of the overhead, archive format header and the like. Gotty don't worry, I'm fine! It doesn't work any other way with him Why are you blind? You've been told 2 times that PAQ already compresses 2 characters! (Reminder, Gotty don't worry, I'm fine!) Well, PAQ already does what you need it to do, it already compresses those 2 characters, only it does it better and faster, without bloating the file and we don't have to convert the file to bit view because PAQ does it itself! Just read: https://en.wikipedia.org/wiki/PAQ http://mattmahoney.net/dc/paq1.pdf You have a lot of time to think about:) ______ Tell me, do you program? Have you really learned any programming language?

I rushed in my last reply which was sloppy You said it well about how conversion loses to compression, but you also answered the question after to state that no one deals with compressing 2 digits since their is no need for it. Which again their is no need for me to answer the puzzle but it was a challenge for others to learn from. Which again if everyone does the same nothing new is being done to get a difference perspective. Your reply is right from a conventional stand point. But why have you not 2nd guessed yourself? Since its not needed as you say. How do you know that you know it all? based on experience. A lot of things are found by accident just fooling around and trying silly things since the obvious is not so obvious. Random to you is not random to me. people with super memory have a structure to remember random numbers of phrases due to association. They may be random for others but not to a few. The computer does not know what random is unless a person puts in what is or is not random which you agree. So to say its impossible due to it being random is like saying it is impossible since it is being ignored. Ignoring something does not make it true truly but temporary to the limitation of the person. And that person puts it in the code which also makes it limited. and then it becomes a standard and then people ignore it since who will argue with the official stance. That does not seem healthy for innovation, since it puts boundaries and discourages. So not everyone says you can not compress random it reached its limit by so many experts and anyone new feels experts know best. That is not healthy for any field of work is what I am trying to get at. words make of break things which have restricted a lot of innovation. Even dictionaries have changed the meaning of words over time. Even something as simple as the word leisure means now to relax with free time but it use to mean to use free time to learn if we go much further back as a quick example. I am not here to change anyone mind I am here to offer another perspective to see things. Since when everyone seems a playing cards from one side it may look wide but from another it is thin. No one has to agree, but just like your tests its a way to get use to thinking and thinking differently to help things. on a side note Tesla likes the #3 and he sid the secret to everything is frequency. When he meaning everything does he mean everything? as for game ABCDEFG which i did not have time to read it last time. but the 8 letters seems to small to answer under 3. 8= you choose 4, 4=2, 2=1. the selection it too small to deal with but needs other methods to get 2. Sure your problem can be solved with 2 but again the odds of it being wrong are greater. LOL but if you only had 2 choices then you have to come up with a theory on why you choose and understand the context of the where it it coming from. not as practical but its a good way to think things through. as for the 256 which was answered last time it might be good to take a guess in the first part to eliminate 2/3 i was trying to take that path than 1/2 which is the sure and steady way to get a shorter answer. it has a probability of being wrong to delay more but when right it has a probability of being fast which was the purpose of the game. if i was right all the time guess and eliminate 2/3 that would be 6 turns if wrong and stuck to it then 13 turns at worse which to be wrong every time the probability would be 50% so within that 50% it would take 8 turns as well. So 5 at best not likely 13 at worst. Which that is how I came to my answer but thought it out now to understand it better to explain the numbers. Unless your percentage is different? SO you are right but I think I gave a better answer in speed which was a risk. You try it at 2/3 off and tell me how fast you get it. answer is 3 and the 2nd one is 200. use the same method which order will you eliminate first is up to you. so what are your odds? I should have puyt more effort last time but as i said i was in a rush and did not have time to think it through. since lotto odds have also a pattern which most dont know, but it looks "Random" to most. The thing is probability to get better odds. Just like if you flip a coin it will be 50% heads or tails. and if you flip it 100 times it should be near 50% heads or tails. Sure it can be heads 100 times but the odds increased despite it always has a 50% change in every flip to be heads despite it it is heads 4 times in a row or 8 times but their are other probability factors in play that most ignore due to what they see at face value which everyone is tricked even most mathematicians. you say "Also don't expect that one day we will invent a smart method to compress random data. " You already do, :) A file is a file. what the program that deals with it determines if it is too "random" for it to handle. So to call a random file not random when you know the code is not fair. it is just like a lock how it is all random but not to the key you have. all locks are the same, its the key that determines if it is in order or not. if anything the compression program is random to use predetermined patterns. Since some programs compress better than others on the same file that is not "random" which is random in how much it is compressed. It is not the files fault it is the program. Or to be fair it is both, and if it does not work the program is not comparable. It is not the 4k files fault that the compression programs do not do a good job which as you said no need to compress 2 character, which is based of programmers preconceived notion. ;) which that is what I am trying to say how everyone is stuck to some degree doing the same thing fear of taking the next step. Everyone is trying to squeeze a dried up lemon and putting more memory and processor into doing so. Its like throwing more money at the problem to need a bigger processor and memory. Again most wont agree which I see xinix (hello) is upset with me and gives you tumbs up all the time. LOL reply is long enough I will read the other game another time. you can try my method if it applies and see if that works. YEP ;) So try to beat 4.0k

Celebrating 1st time ever random files compressed a little smaller and reconstructs exact same ( no further details will be provided until separate encoder decoder done ) Subject to eventual final confirmation with separate encoder decoder over 2 clean PCs

Random files? You mean: more? I thought it was jut a zstd file. So you have different files, right? That's good! I just wanted to ask you to double check the results with real random files. xinix also posted one. You also referred to random.org files in your earlier posts, so I believe you have now some test files with you. So what are the (preliminary?) results? What are we celebrating?

Sportman's provided random files ( pls leave any further requests for technical details for when separate encoder decoder done. None will be responded at this time)

Are these inputs random files?

Hi , I have made good progress coding done compressor and have now greatly made simpler decompressor At moment decompressor works direct from compressed internal variables decompressed exact same. There is only work to separate into encoder decoder. But do not underestimate the amount of 'mundane' tedious work involved .Likely will be a week's time to able do as requested. But we can and should already be celebrating. ( now can handle any size input also ) LawCounsels