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[silverspawn]

Also unrelated (but brilliant):

[silverspawn]

Still nowhere near a solution

I think the causal diagram here is

Number Distribution -> P1 Action
|
 -----------------------> P2 Action

So by observing the number distribution, P2 can learn up to maximum entropy (~5,5 bits per box) about P1's Action. But this itself is completely useless because we're not graded on how similar or different P1's and P2's actions look. I mean, you could also just fix them in advance.

I don't think it is possible, given this diagram, to learn more about the number distribution. Therefore, I think that P(P2 found the right number) = 0.5. And ditto for any other player. I think all 100 players will 0.5 chance of finding their number.

If this is true, the trick is contained entirely in making the successes correlated. P(P2 found her number | P1 found her number) > 0.5. Not completely sure about this, though.

[silverspawn]

I was initially going to say something like "at some point while she's playing, P2's distribution over the remaining numbers has to change" but that just seems impossible.

[silverspawn]

An equivalent way to put this, if you think in terms of communication bandwith, is that, although the true bandwidth is 0, you can pretend as if it's 1 per game. Like there's an equivalent game in which each prisoner is told that the one before her finished successfully. That's 1 bit which contains information about the number distribution. Then the question is, how do you use that maximally well.

[silverspawn]

Unrelated, extremely interesting but also hard to follow video. Among other things, there's an argument here that observing the computational properties of vision is itself enough to show that the brain uses holistic computations of some kind because otherwise the tasks are too algorithmically complex. I foresee a future in which I spend a lot of time dealing with this.

In the end, this kind of thing may be half or even more than half of the story. Ultimately, the main purpose of talking about consciousness and showing that digital computers can have it is to prove that human brains do holistic computation. But you can also prove it directly. In practice, doing both makes sense.

[silverspawn]

*can't

[silverspawn]

This solution is going to blow my mind when I finally derive or read about it

Another attempt:

Thinking about how to model this. I'd like to choose objects for players such that I can afterward assume a uniform distribution over F (the space of all orders of boxes). In other words, first the prisoners choose their strategy, which I model as functions that take all information they may depend on as parameters, and *then* the distribution over F is uniformly and independently chosen.

So I'm still just in the step of figuring out how P2 should play. Looking at each choice of P2 separately. The first cannot depend on any aspect of the real function f in F (i.e., how what the boxes look like). So the 0th box that P2 chooses (counting from 0 will make more sense here) is just a constant B_0 in N. Consequently, the probability of hitting the right box is exactly 1/100.

The next one however (box #1) can depend on the number in box 0. So it's a function B_1 : N -> N. And the box after that is a function B_2 : N^2 -> N, the next a function N^3  -> N and so forth; the n-th choice is a function N^n -> N.

So it SHOULD be easy to find a B_1 that has probability better than 1/100... since one probably exists (seems unlikely that we can only do better later on) and also there aren't that many choices.

[silverspawn]

Well actually there are 99^100 choices but well

[silverspawn]

And ofc this can't be looked at in a vacuum; we assume that P1 before succeeded.

[silverspawn]

I failed. The problem was frustrating me too much, and in particular, I kept finding "proofs" that it's impossible, so I looked up the solution.

The biggest mistake I made was to think we can assume wlog that the first prisoner just opens boxes 1-2. Because if she does, even case n=4 is unsolvable. But it seemed like that assumption had to be ok since the first prisoner has no information and hence we can just relable the boxes afterward...

I even had permutations in an abstract algebra class and we covered the cycle notation.

[silverspawn]

Also the solution feels unsatisfying. Like it feels as if I'm yet to find the deeper flaw in my reasoning

[silverspawn]

Right now it actually feels like magic

[silverspawn]

Like my main hangup was "you can't make the strategy of P2 correlated with that of P1 because the task of finding number #1 is so different from the task of finding #2" and I still don't feel like I understand why that was wrong.

[MiX]

Did you solve it for n = 4?

That one felt much easier, since you can have P1 check different boxes as their second peek depending on what number they saw in the first one, and since the second player has to assume P1 found the 1, they have information on where that 1 is. I could only extrapolate to any n due to reading the spoilers: even if it felt intuitive that it was optimal, there's no way I could prove the probability was >0.3.

[silverspawn]

No, I failed to prove it for n=4 because I assumed player 1 can just pick boxes 1 and 2. And if P1 does that, you can show that P2 can't get above 2/3, which is what you get by picking the other two boxes. And that's equivalent to the 50/99 that P2 gets by picking the 50 other boxes for n=100.

[faust]

Did you solve it for n = 4?

That one felt much easier, since you can have P1 check different boxes as their second peek depending on what number they saw in the first one, and since the second player has to assume P1 found the 1, they have information on where that 1 is. I could only extrapolate to any n due to reading the spoilers: even if it felt intuitive that it was optimal, there's no way I could prove the probability was >0.3.

The main trick here I think is not that P2 will learn information on where #1 is - that would be true even if you had P1 check just boxes 1 and 2 - but that P2 learns information on where #2 is as well! (Since #2 is in box 1 if and only if #1 is in box 2)

Indeed you could view the optimal strategy as giving up some of the information on where #1 is in favor of more useful information.

[silverspawn]

It really annoys me when anyone says silly things like the best unit in broodwar is the defiler or mutalisk or zergling.

The best unit in broodwar is the PROBE. Like if you made a list of best units other than workers, protoss doesn't get a spot in the top 5 and yet they're about as good as the other two races? Why you think that is? BECAUSE OF THE PROBE. IT'S ENTIRELY BECAUSE OF THE PROBE. The entire race is structured around that. It mines faster, makes macro easier, means you don't need to take workers off from mining to build stuff. It's why the race gets worker leads in every matchup, has more supply, and can expand so easily.

No-one else ever makes this point

[silverspawn]

oh and it's also PROMPT

[silverspawn]

like every time protoss builds a building they get the equivalent of ~50-100 minerals compared to terran. That probably has something to do with why they are competitive despite the fact that all their units look weaker on paper.

[silverspawn]

Here is the video about the puzzle btw. It's from the channel faust linked for the complex numbers earlier. Really good stuff.

[silverspawn]

[Awaclus]

oh and it's also PROMPT

*ALSO PROMPT

[silverspawn]

So according to the Veritasium video, the following things may or may not help with aging:

- 0: don't get DNA damage. Use sunscreen etc.
- 1: eat less; specifically less protein
- 2: get really exhausted from e.g. working out
- 3: be uncomfortably cold or uncomfortably hot

If AGI requires a paradigm shift it could take who knows how long so this is again relevant.

[silverspawn]

#0 seems easy enough. #3 I think I'm gonna pass on. If I had a sauna in my home, sure, but alas.

#1 and #2 are interesting though. I did resolve to try to only eat within a 4 hour window each day. Which is not that big of a change since I was kind of doing something like that anyway, but not consequently.

And for #2, yesterday I did both workout sessions in one instead of spaced out. That's probably still not as intense as you should get though.

[silverspawn]

I've also heard good things about calorie restriction elsewhere, and one meal a day doesn't seem like it's bad for you in any way. I don't in fact get super hungry.