Conquering Difficulties Question

[thedoctar]

I recently solved problems 895 and 910, and I only need to solve a 90% difficulty and 100% difficulty problem to get the Conquering Difficulties award.

Currently, problem 895 is rated 90%, but has the (Not yet finalised) caveat. Problem 910 doesn't have a difficulty rating.

However, on the Conquering Difficulties award, it says I have done 19/20 required problems, but on my progress page, I don't have any 90% problems solved.

So how does this award work with new problems? Does it consider the (Not yet finalised) difficulties? What happens if I solve a different 100% problem, but problem 895's difficulty changes? Will I be unawarded the award?

How long does it take for the difficulties to be locked in? How long will it take for problem 910 to have a difficulty? If it has a 100% (Not yet finalised) difficulty, will I get the award? If the difficulty changes, will I be unawarded?

Sorry for all the questions, but I want the award, but solving 90% and 100% problems isn't easy and I'm also a bit busy.

[philiplu]

If you're trying for the Conquering Difficulties award, then it makes more sense to solve finalized problems in the archives.

Only problems with finalized difficulties participate in any of the difficulty-based awards, so none of the 10 most recent problems, nor older problems that haven't hit 100 solvers yet, will qualify. Once a problem hits 100 solvers and enters the archives, then it is considered for the awards (pretty sure about that).

It'll probably take the better part of a year for 895 to hit 100 solvers (from the time the problem was released). I suspect it'll be at 95% or 100% by the time that happens.

[thedoctar]

Thanks for the advice, and for the estimate for 895 difficulty. Good to hear I solved a hard problem! But sounds like it won't help me for the award .

I'm really surprised it takes that long for 100 solvers!

One more question: when I sort problems by difficulty in the archive, I get a definite order for problems within the same class that isn't based on when they were released. E.g. the top problem when ordering by descending order by difficulty is #763, the amoeba problem.

Is this the “hardest” PE problem, harder than all the other 100% ones? Or is the ordering just random?

[philiplu]

I have no knowledge of the inner workings of PE; I'm just a long-time participant divining how the black box works from the outside. So take what I say with a large grain of salt.

As for length of time for hard problems to hit 100 solvers, think of it like this. If you go to the Statistics page and click "Show active members only" in the upper-right, you'll see there are about 100K members who've solved at least one problem in the past year. Click into the Problems page, and you'll see about 1750 of those active members have solved at least 200 problems. So there aren't that many people actively trying to solve a bunch of problems, and fewer of those will try to tackle the hardest problems. Not too surprising, then, that the hardest problems take (well over) a year to hit 100 solvers.

When you sort by descending difficulty, then, in some fashion you are getting the sorted list of most difficult problems, by whatever metrics are going on behind the scenes. You can see that time to 100 solvers is important but not absolute. The first three problems in descending difficulty order are 763 (3D Amoebas), 780 (Toriangulations) and 566 (Cake Icing Puzzle), which took 2 years 17 weeks, 1y51w, and 2y38w respectively. So something else other than time to 100 solvers is in play. There are posts in the chat forum mentioning the slope of the time-vs-solvers curve being important as well, but the PE team is, I think, intentionally leaving things vague.

I've noticed that difficulties will, rarely, be adjusted well after a problem hits 100 solvers.

OK, I got curious while typing this, so I plotted the solve times for those 3 problems. X-axis is time in weeks, Y-axis is number of solvers. Even though 566 took the longest to get to 100 solvers, you can see it got close to 100 the fastest of the three, so the ordering makes sense if you look at time to 95 solvers. Interesting.

[thedoctar]

That's a super interesting plot! I thought about it in the shower, because I can never stop thinking, and I think if there was some kind of decay law for probability of someone solving vs time (or 1/(time to solve) vs nth solver, so 2^-n means next solver takes twice as long as previous), you'd have different behaviours for the following reasons:

1. People competing in the Eulerians league table would mean that that first 50 solvers gives you an idea for how difficult it is for people who have a lot of knowledge and experience in maths and programming, who actively practice that skill and use it competitively
2. I'm guessing problems whilst they're in the recent page get more publicity, so more likely for people to solve it
3. Once a recent problem goes into the archive, I'm guessing that far fewer people are willing to solve it, as they guess it's super hard, plus the only reward is having your name in the top 100 list (although I think it's still a pretty good reward).

Based on your graph, it seems that the amoeba problem (763) was less difficult than 780 for the super-competitive Eulerians, but more difficult for the less-adept non-Eulerians. Since the difficulty system is aimed at the “general public” *cough cough*, I guess the 50 to 100 solvers are important. There might also be other effects like holiday period (where I guess the number of solvers increase, plus dedicated solve time).

I also wonder if there is some kind of normalisation effect based on number of active solvers. PE probably has own dedicated analytics service for that? It would be interesting if they observed seasonal/periodic effects like Xmas time.

[Swistakk]

If you are still hunting for 100% and 90% problems from the archive, then for 100% I would recommend going for 502, 559, 696 that are in my opinion visibly out of their place. In 95% set I see some outliers too, but curiously enough I don't see any as easy as these in 90% set. Personally I found 514, 677 very easy, but that was more "if you know, you know" kind. Or you may snipe both difficulties at once by going for a certain peculiar combo of two basically identical problems, one from 100% and one from 90% set, though I will not directly spoil which of them I have on mind (which are still on the easier side of these difficulties). Your mileage may vary, obviously