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I've played this game before, but it has been a very long time, and I've never really tried speedrunning it, so forgive me if this is completely off base.
A minor question regarding the collection of the two small yellow cube pieces at 45:15 IGT, you spend several seconds running over these three pillars. Is there no way to use a perspective change to access the far right pillar from the far left one? The ladder attached to the third pillar is too far down, and the jump is probably much to far to make it to the 4th structure here, but is it possible somehow to do a ledge grab or land at the exact correct time or something during a perspective shift?
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Instead of solving explicitly for x, it's easier to square x + 1/x to get x^2 + 1/x^2 + 2 = 293. Notice that the goal formula reduces to x - 1/x, and if you square that you get x^2 + 1/x^2 - 2, which is 293 - 4 = 289 from our above formula. Then just take the square root of 289 = 17.
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Am I missing something? Why is this guy being permabanned? Reuploading isn't against the rules. And I can't know for sure because the youtube accounts have been deleted, but from what I've read in the rest of the thread, it doesn't seem the guy was making a mockery of anything?
From a more constructive viewpoint, if this behavior is not desirable, then should the licensing terms of the TASes be changed?
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This problem was given to me via a Discord help chat, and I thought it was an interesting problem. The labels on the hypotenuses are referring to the entire hypotenuse. The student who gave this problem fell into a bit of a trap, and I was unable to convince him that the easy method was the intended one.
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Apropos of nothing, anyone know an approximate closed form for: 2.405565034289246
This comes from a numeric solution, a, to minimizing the squared error between (1 + erf(x))/2 and 1/(1 + exp(-a x))
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If you have two standard d12s, the probability distribution of the total value of rolling 2 of them is a triangular pmf. How many ways can you relabel the numbers on the d12s to get back the exact same distribution? (subject to the restriction that the values must be positive integers). Is there a generalizeable way find this do this for rolling n dice with k sides?
An example for d6s, is the standard dice both have faces {1, 2, 3, 4, 5, 6}, while the (unique) way of relabeling them such that the analogous situation is true is the first dice is labeled {1, 2, 2, 3, 3, 4} and the second is {1, 3, 4, 5, 6, 8}.
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Myers-Briggs is more or less psuedoscience. There are more nuanced personality tests that are used in scientific disciplines (such as Big 5), but boiling down people to 16 "types" is a bit of a fools errand. The categories are vague because they must be in order to be even slightly believable (like horoscopes).
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This winds up being equivalent to finding the intersection of a parabola with an ellipse*. If you straightforwardly attempt to solve the resulting quartic, you don't get anything that looks reasonable. The quartic I arrived at is 1396 u4 + 296 u3 - 1744 u2 - 192 u + 480 = 0, where u2 = y.
I assume there is a tricky change of variables that makes this possible/reasonable to get a closed form solution, but a numerical approximation is y = 0.9453, and x = 5.0547.
I'll circle back to this when I'm less busy. Seems that if I can rotate the parabola in order to eliminate the + 4u, I might be able to perform a substitution on the ellipse that might make it more amenable to manual calculation. However, I'm not looking forward to dealing with an off-axis ellipse, and it might not wind up being any less complicated.
* the parabola is 37u2 + 4u - 24, and the ellipse is 4 sqrt(6 - u2)
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I don't believe that there is a simpler way to do this, I did largely the same approach, but you managed to cut out some of the faffing about I did with x = f(u), y = u. Great job.
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The Riddler is dead, long live the Fiddler.
This week's extra credit Fiddler problem has been giving me some issues. I will restate the entire problem first, then explain where I am stuck in spoilers.
With a little bit of effort you can find the values for N=1 and N=2 to be 35 and 9900 relatively easily or nearly instantaneously via a computer search, I won't bother doing much exposition on this part.
Here are my thoughts on the final part for arbitrary N:
If you examine the form of the solution for N=2, you'll find that all of the minuses become pluses other than the one targeting the 2, and similarly, all of the divisions become multiplications other than the one targeting 1. So this would seem to suggest that the pattern should be find some combination of the additions that sacrifice the 2 to the minus and the 1 to the division, leaving behind a grand total of 8N^2 + 6n - 4 to be split among 2n terms as evenly as possible to maximize the multiplication at the end. This gives (4N^2 + 3N - 2)/N on average, or between 4N+2 and 4N+3, approaching 4N+3 as N gets large. Using this technique it's possible to, instead of enumerating permutations of digits operations and orders, instead simply enumerate partitions of these integers into exactly 2n terms, and then choose some set of operations that makes it work. Using this approach you can rapidly find N=3, N=4 takes a few hours, but N=5 seem a little bit out of reach computationally still.The reason I want to enumerate the case for N=5, is because I suspect that it might become necessary to also sacrifice 3 once the number of minus operations becomes sufficiently large, in order to split the remaining values more evenly into the multiplication bins, finding if this becomes required, and if so exactly when would be required to begin categorizing a general solution.
Using this technique, the best value for N=3 is (8 - (2 - 3 - 7))/(((1/(6+9))/(5+10)) * (4+11) * 12 * 13 = 13 * 12 * 15^3 * 16= 8424000, and for N=4 it is 17 * 16 * 15 * (11 + 9) * (12 + 8)/((1/(13+7) /(14+6)/(10 - (2 - 5 - 4 - 3))) = 17 * 16 * 15 * 20^5 = 13056000000. For N=5, the search is yet ongoing but a candidate solution is (found by hand): 21 * 20 * 19 * 18 * (17 + 7) * (16 + 9) / ((1 / (15 + 10))/(14 + 11)/(13 + 12)/(8 - (2 - 6 - 5 - 4 - 3)) = 21 * 20 * 19 * 18 * 25^4 * 24^2 = 32319000000000.
One way to measure the "badness" of a particular solution is to find the squared deviation for each multiplication term from the average term, then divide by the number of terms. This roughly captures (or maybe exactly captures, I honestly haven't bothered to try to give a proof either way) how far away the solution is from an unobtainable upper bound. For N=3, this value 1.889, for N=4, it's 4. The candidate solution above has N=5 at 8.8, with just the 18 term at badness 21.16. So attempting to eliminate this term we can rewrite with also a -3 as well and get (not bothering to rewrite *s and /s and +s as -s), 21 * 20 * 19 * (18 + 5) * (17 + 6) * (16 + 7) * (15 + 8) * (14 + 9) * (11 + 10 + 4 - 3) * (13 + 12 - 2) = 21 * 20 * 19 * 23^6 * 22 = 25989180672840, which while this solution only has a badness of 2.95, the fact that there were 3 additional units to work with seems to easily make up for the relatively high badness of the N=5 candidate.If we assume that -3 will never be bigger, then we can come up with product(3N+3..4N+1) * ceil(S/(N+1))^(S mod N+1) * floor(S/(N+1))^(N+1-S mod N+1) where S = sum(3..3N+2) - 2, but I'm still not entirely convinced this is optimal for large enough N.
EDIT: In fact, consider two numbers that sum to 1001, the largest pair of multipliers are 501 * 500 in this case, if we subtract 3 from the total we have 499 * 499, now unbalancing the first, we have to go all of the way down to 461 * 540 before the unbalanced but larger value becomes larger than the balanced smaller one. This is a total "badness" of 3120.5, which is about 3 * 1001. So it seems like the total badness will have to reach very large levels in order to justify removing the 3, roughly 3 times one of the terms. However, they do seem to become bad enough. This would suggest at upper limit where that things become bad enough at N=49 to force using the -3, and perhaps it's better a few (several?) values before.
EDIT: A much more careful analysis of the badness for these two cases suggests a crossover at around n=19, and an analogous formula for the sac 2 and 3 case suggests that the crossover value is exactly n=21: (21, 1858561114113223590154513972642442186649575491096420050912878442728057261260800000, 1861170011681117541222317931520613643629245091916594782109445096734720000000000000, False)
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Is there a nice solution to this? It feels like the coupon collector's problem, so I would assume that the answer is 1/e in the limit, but every approach I've considered and that I know to be correct has horrible dependent probability mess.
I'm not completely certain if this approach is kosher but it gives me the answer I'm expecting, and I don't care enough to remind myself of the exact rules about expectations to justify it properly, but consider N squares, the expected number of black squares at the end of the first turn is p1 = 1, then it's p2 = p1 + (1 - 1/N) = 2 - 1/N for the second turn, and p3 = p2 + (1 - p2/N), and so on. Then it's a matter of solving this recursion.
p3 = p2 + (1 - p2/N) = 2 - 1/N + (1 - (2 - 1/N))/N) = 2 - 1/N + 1 - 2/N + 1/N^2) = 3 - 3/N + 1/N^2
p4 = p3 + (1 - p3/N) = 3 - 3/N + 1/N^2 + (1 - (3 - 3/N + 1/N^2)/N) = 3 - 3/N + 1/N^2 + 1 - 3/N + 3/N^2 - 1/N^3 = 4 - 6/N + 4/N^2 - 1/N^3
I could be wrong, but this seems like a Pascal's triangle situation. And again, not caring enough to justify this, let's just move on and assume I'm correct. lol. If we divide both sides by N, then subtract 1 from both sides we will get 1 - pN/N = (1 - 1/N)^N which is e^-1 as N gets large.
pN/N is the ratio of black squares to all squares, so 1 - pN/N is the exact value we were trying to find.
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So to find critical points of this function we just need to take the derivative. This is (x^2 - x - 6)e^x which factors into (x-3)(x+2)e^x. Set equal to zero and we have critical values of m at y = 0 (from e^x at negative infinity), y=7e^-2 (from the (x+2)), and y = -3e^3 (from the (x-3)). So this function has zero real roots for m < -3e^3, exactly one at m = -3e^3, two on the range -3e^3 < m <= 0, three on the range 0 < m < 7e^-2, two at m = 7e^-2, and finally one for m > 7e^-2.
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MLE is the intended solution, or rather, conditioning on the likelihood. If you collect all of the likelihoods, normalize them as a pmf, then evaluate the expectation value of the given hypergeometric distribution, 5/14 is indeed the result you get for an arbitrary bag of fixed size N, then you'll find for any valid choice of N > 13. Proving this is somewhat tricky though and it involves some more obscure summations involving binomial coefficients.
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So the initial probability that a blue or yellow tile is drawn is an unknown. Assuming the bag initially has 50% blue and 50% red tiles is not justified.
As you may have observed, the only assumption you need, once you condition upon the likelihood of the observation of the initial draw, is that the probability of any tile being drawn (independent of its color) is uniform.
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I have a bag of tiles. The tiles can be either blue or yellow, but you do not know how many are blue and how many are yellow. You draw 12 tiles without replacement: 4 blue and 8 yellow.
Is the next tile you draw more likely to be blue or yellow?
Does the answer depend on the number of tiles originally in the bag?
What is the chance that the next tile you draw is blue?
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