Hi, this is part 3 of posting your Math Problems in this thread, and I will try to solve them.
The solutions to #2 [https://lichess.org/forum/off-topic-discussion/post-your-math-problems-here-2] are also given. I wasn't able to solve them in the thread as I was pretty busy with stuff, and it has become archived. Also, it had become pretty huge, so here we are. I've left out the "1+1=?" questions. If you have asked a "1+1=?" question, the answer is 3. Shoutout to NM @Bixit00 ;)
Also, sorry for the pings :(
I'm also leaving out Question 2, asked by @Samboy2023, as the solution has been given in #6. Questions have been answered in reverse order (sorry!), but I have tried to answer all. Moreover, I will try to keep this post as concise as possible, so please notify me for any mistakes or anything I've misunderstood/overlooked. Feel free to DM for clarifications or ask in this thread.
They were all interesting questions, so this might take some time lmao :)
I couldn't solve number 24 and 15, so feel free to try it out guys. Btw, 16 is a well-known unprovable (even now) conjecture
74) @Mulafi Fractional powers are calculated by taking the numerator as the power and then you calculate the denominator's root of the result. For example, 7^(4/5) = the fifth root of 7^4. To calculate decimal powers, you need to convert it to a fraction. 4^2.7 = 4^(27/10) = the tenth root of 4^27. Intuitively, even decimal powers of negative numbers shouldn't exist as the answer's imaginary. A trivial example is (-4)^(1/2) which gives 2i, where i is the square root of -1.
50) @tcip I would say the answer is 8, where the pawns of all pairs two adjacent files en passant each other, one at a time. There are four pairs of files, so 4 x 2 = 8. For instance, in the g- and h-files, 1. h4 g5 2. h5 and 3. hxg6, and 1...h5 2. g4 h4 and 3...hxg3. You feel me? The pawns aren't blocking each other.
47) @SDawud Skipping a few steps, the given series can be written as the summation from 1 to 40, [(4k - 3)^2 + (4k - 1)]. Expanding, we get: The summation from 1 to 40, (16k^2 - 20k + 8). Simplifying again,
= (16 x sum[1 to 40] k^2) - (20 x sum[1 to 40] k) + (40 x 8). Now, sum[1 to n]k = n(n+1)/2, and sum[1 to n]k^2 = n(n+1)(2n+1)/6, so we get 338,160 at the end.
45) @PenguinWhack271828e Nice question, for the trivial case I feel it would be all knights on a certain color, so 32 knights. For a queen, placing it at a square where its reach is least is wise, so we sould put it in a corner. Then, leaving the adjacent squares (2) free, fill in the other (non-attacked) diagonals with knights. For the rooks, I would place them at corner adjacents, and fill the other diagonals with knights, same way. Similar approach for the king on the hill, leave some squares empty and fill the rest with knights. Similar with bishops too, place them at a corner and fill the non-attacking diagonals with knights.
43) @To_be_is_to_do It's a Mersenne prime with over 41 million digits. I have forgotten the exact value, so please check the internet, sorry :(
41) @ATREYAhyper 256 by trial and error. You can do it this way too with logarithms, it makes it easier: y ln y = 2048 ln 2. Let y = 2^x, so x.2^x = 2048, by a simpler trial and error we get x = 8, so y = 256.
40) @PenguinWhack271828e The counting of the odometer is in base 9, so converting base-9 to base-10 (decimal), we get 11,254 km. Sum of digits = 13. Excluding 4, we define a generative function f(x) = (sum[1 to 14 and neq 4]x^n]^5, and now we have to find the coefficient of x^13. Or, let a set = {1, 2, 3, 5, 6, ..., 13} We can manually count the unordered combinations lol so this will take some time, but by brute calculations you should get 32. The third part is simpler, substitute B = 32 and use the distance formula.
34) @richniftyrabbit -1/3.
32) @Ar4Asd1_Chess Math breaks down here lol. I can't make you happy, but can I give you a cookie? :D
24) @candy_0930 Yeah I think I remember this one it had something Euler-ish. Y sides implies Y midpoints (corners here) = k (let). One-stroke implies polygons have some relation/connection. Let G be the reqd midpoint graph, so Y times (2X - 1), ummmmm ok I'll have to think about this, nice
23) @limodet Omg yes, I recall that standard case, definite integral from 0 to 1 of ln x/(x^n + 1) is -[pi/(n sin(pi/n))]^2, only from 0 to 1 and n>1, so it should be easy by substituting n = 5. Silly me tried to find out the indefinite hehe and I ended up wasting 15 minutes lol
22) @PenguinWhack271828e Total 138 moves, or 69 moves by white. Taking the probability of the moving a knight, diagonal and straight move for each move to be a, b and c respectively, set up a + b + c = 69. Now you've gotta substitute reasonable values for a, b and c. Once you've done this, you can set up a(1) + b(sqrt2) + c(sqrt5) = required distance.
21) @PenguinWhack271828e Probability of an existing number to be generated = (n+1)/(2n+1). Let k10 -> the stuff at 10, so 3 | k10 and k10 congruent to 0 (mod 3). Remember the Chinese Restaurant sum? Same method. In the end, you'll see that anything's probability of selection at n+1 is proportional to how often it has been selected before. Thus, number 8 on move 20 will be, by applying the derived formula from CRP, 1/[(20 - 1) + 1], so it's 1/20.
18) @ajfang nice one, and sorry for those facts hehe :|
16) @AsDaGo Goldbach conjecture lol bruhhh it's unsolvable :(
15) @HuaTianXiangQi I don't think there's a proof for that via number theory. However, it has been proved by brute calculation.
11) NM @Bixit00 welp hehe :|*discreetly opens calculator*
7) @Damkiller25 Umm that's a bit vague question, could you describe it a bit?
8) @Damkiller25 In these problems you should just represent it in a 3D-coordinate plane, which makes things a ton easier. Once you've done that, you need the distance formula to calculate points, the heights and thus whatever you need. Keep in mind that the base is an equilateral triangle, and you should be able to solve for the S, D, and E pretty good.
Alright, so that was all the solutions. Looking forward for new ones :)
So y'all can post your math problems here
123*104*x+787=12344556. x=?
@PrashantSrikanth said in #3:
> 123*104*x+787=12344556. x=?
Simplifying, we get:
12792x + 787 = 12344556
or, 12782x = 12343769 [transposing]
or, x = approximately 965 (964.96) [dividing both sides by 12782]
@ForumMathSolver said in #4:
> Simplifying, we get:
> 12792x + 787 = 12344556
> or, 12782x = 12343769 [transposing]
> or, x = approximately 965 (964.96) [dividing both sides by 12782]
Great job!
@ForumMathSolver
If 3 to the x - 1 is 2,
What is 9 to the x + 1 power?
Yeah, I'm guessing this is pretty simple for you.
@Anfdeey said in #6:
> @ForumMathSolver
> If 3 to the x - 1 is 2,
> What is 9 to the x + 1 power?
We have: 3^(x - 1) = 2
Squaring both sides, we get: 3^2(x - 1) = 2^2
Or, 9^(x - 1) = 4
Multiplying both sides by 9^2 = 81, we get:
9^(x - 1) x 9^2 = 4 x 81
Thus, 9^(x + 1) = 324.
I hope this helped man :)
Bro qualified harvard and plays chess after retirement and helps people doing homework,what a great guy we need more people like him in lichess
@ForumMathSolver said in #1:
> Hi, this is part 3 of posting your Math Problems in this thread, and I will try to solve them.
> The solutions to #2 [lichess.org/forum/off-topic-discussion/post-your-math-problems-here-2] are also given. I wasn't able to solve them in the thread as I was pretty busy with stuff, and it has become archived. Also, it had become pretty huge, so here we are. I've left out the "1+1=?" questions. If you have asked a "1+1=?" question, the answer is 3. Shoutout to NM @Bixit00 ;)
> Also, sorry for the pings :(
Use the Conway Doomsday Algorithm to find out which day of the week August 5th, 1874 fell on.
@ForumMathSolver said in #1:
> I recall that standard case, definite integral from 0 to 1 of ln x/(x^n + 1) is -[pi/(n sin(pi/n))]^2, only from 0 to 1 and n>1
I'm afraid that this answer is wrong. I knew it when I calculated the integral with my hand. Your ans is even smaller than I_2= \int_{0}^1 {lnx/(1-x^5) dx}
According to Maple calculator, the numerical results show that
The ans - Your ans ≈0.1647031594
and that I_2 - Your ans ≈0.091978966
I should say SORRY because I also find the summation of the series may have no simplified expression. So this problem is not very good, I didn't do it until I see your reply
However, I think you do know the right way to solve the integral for your answer is related to the Series I obtained, but you made some mistakes.
One of your mistakes can be misusing the identity
π^2/sin^2(πz)=∑ (z+m)^-2 where z is not an rational integer and m runs over all integers (not only those non negative ones) and this is why your ans smaller than I_2.
Additional comments on this question. To expanding the integral as series we need to apply the dominated convergence thm to communicate the integral and summation. I don't know if you're aware
Anyway thanks for paying attention to my question, my suggestion for you is that you can use some maths software to check your ans and don't worry too much about making mistakes
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