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Is there any sort of hard upper limit to the quality of random numbers that we can produce with ONLY our brain? Are there any methods to increase the quality of random numbers we can produce?
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Humans are horrible at generating random data. Don't even bother.
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There isn't a *mathematical* reason humans would have a limit on the kinds of random-ness they could generate, they should be able to generate random-ness as well as physics says that you can at all, and that is a physics question (e.g. about the nature of radioactive decay and such).

The deeper mathematical question is what do you even mean by a *random number*? There is no obvious well-defined concept as "pick a random number". If you want to pick *uniformly* (i.e. every number equally likely) then that would be impossible for a human to do because there is no way to create a uniform distribution over the 'numbers' mathematically at all.  So what is mathematically possible depends on what you mean by *random*.
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return 4;
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As other commenters pointed out, humans are terrible at generating (pseudo-)random numbers directly, but really so are computers if you don't give them a fancy algorithm.

Consider this algorithm the following algorithm for a human D6:
Think of any 5 digit number.
Find the remainder when dividing by 7.
If it is 0 try again.

Though we are good at identifying patterns and the distribution of 5 digit numbers we generate is certainly not uniform, most of that is lost since (to our minds) numbers modulo 7 do not have a clear pattern.

Thus, this creates a decent pseudo-random number generator.

It can be improved by thinking of numbers with even more digits and dividing by larger factors. The only limit is your memory and arithmetic ability!
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If I were to try to do that I would just compute iterations of some decently simple dynamical systems, a few number of steps can sometime be reasonable to
Compute mentally while still providing a reasonable chaotic element to the output number
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this is not a math question.
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English is not my native language, so I am at loss considering the question... Can please someone tell me (possibly ELI5) what op means?
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For a human being of exceptional skill at mental arithmetic, I think the Blum Blum Shub generator would be feasible and pretty decent. Ultimately, I think the infeasibility of mental arithmetic is going to show up as the limit far before any formal computational complexity theoretic results kick in.
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As others of said , the quality would be low. Partly because humans are terrible at understanding randomness and probability. Here's a simple though experiment.  

If you flip a fair coin 10 times in a row once and only once and get 10 heads, that coin probably isn't fair and random, but instead biased.  

If however you flip the coin 1000 times in a row, laws of probability say there will be like a 99% chance that a string of 10 heads in a row occurs somewhere. A memoryless event (outcome independent of the previous outcome) with probability 1/2 demand it.

Now if you DONT find 10 heads in a row (or 10 tails in a row) , that is now and indication that the coins are now biased and lost their memoryless independence property.

Now ask our human to create a string of 1000 random coin flips like tossing a fair coin would do.  I would bet you a large sum of money that the human would go out his way to avoid calling 10 heads (or tails) in a row because that goes against the common layman understanding of randomness.   In fact a lot  of runs (e.g. 9 tails in a row) would have lower distribution than expected mathematically. But we just established this not what a uniform unbiased memoryless coin would do.

Humans have memory, especially unconscious memory and beliefs and bias, so it is very hard for them to mimic a memory unbiased random entity.

This reminds me when I was a smartass youngster and there was a huge lottery prize and my parents planned on buying multiple tickets and asked me to pick a random selection of 7 numbers from 40 numbers. I chose 1,2,3,4,5,6,7.   My parents angrily complained that wasn't  "random" and could never connive them that my selection had an equal chance of occurring as  a 'random' selection of 3,15,9,24,29,41,33.
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There is a Website somewhere that will win against you in either Rock Paper Siscors, or guess a Number like 90% of times after the first 5-10 rounds. It does that by correstly predicting your future actions from your past actions because you Human suck so much. Google around and you will find it.
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