> Sometimes they can have literally 1 legal move and still take 10 seconds to find it.

Perhaps a nit-pick: they need to discover the legal move, and discover that no other moves are possible, right? As a rather basic chess player myself, I can imagine I might spend some time on this depending on the situation.

I think the idea is that an experienced player would recognise this situation immediately (usually there is only one legal move because you are in check, or there is only one move that doesn't put you in check), or even ahead of time (knowing that their opponent's move brings them in check) and just play the only possible move straight away.

Of course a beginner would take some time to spot this, but it's unlikely you would confuse a beginner for a cheater, since a beginner will likely make many sub-optimal moves and spend lots of time thinking in general.

Depending on the situation, the chess website/app might show you that there is only 1 left. Already, when in check, you can't play a move that doesn't stop or cancel that check.

Sounds like bullet chess is the only answer

I don't want to advertise cheating or give any info about cheating, but a "good" cheater can also cheat in bullet and may be even go more undetected if they were to mix moves well.

Note: I have not cheated myself, but I can definitely see a way it can be done. I'm not sure if it would be good of me to describe the process of course... Just think what input you can have and what output you can get if you were to do this programmatically and you can very well imagine if it's possible, there are also existing tools for that. You don't have to open a chess engine in another window and manually do the movements, if you know how to script.

In bullet, I think may be, you could technically even use something to do "anti blundering", meaning you will blunder a lot less, because engine will just check whether it would be an obvious blunder, and block your move. May be you just allow few blunders, and I imagine it would be undetected. Sorry, again for brain storming about that. It is fascinating topic though. Engine could be running on a lower depth and it could be more sort of positional engine that does not do magical engine moves, but is trained on human players and using neural network mostly. Maybe it will just help you do theory openings. And you won't be able to charge anyone for cheating for following opening theory.

At the end of the day it's akin to the "true randomness" problem. How do you prove a random numbers generator isn't truly, or fully random? Or that chess moves are "truly human", and not computed by an engine. You can only approach this probabilistically.

I also think the cheating detection algorithms can be beated, and I believe I could do that. Why do they still serve their purpose, more often than not?

To me, it's inherently linked to the very nature of online cheating. It's essentially a futile, nonsensical activity. The only gratification is an illusion of intellectual superiority, whose worthlessness is so transparent that it can only attract people who don't get to experience the sense of intellectual superiority pretty much anywhere else. As harsh as it may sound, your average cheater is rather stupid. That's why it isn't really difficult to catch 90% of them.

I don't rule out there are some cheaters who do it out of intellectual curiosity, but that would be a statistical outlier.

My ideas based on what you've said now are simply don't play ranked matches online, just develop your skills against humans and engines.

For real ranked matches, participants must have 360 webcams and so on showing they aren't cheating