Math and Logic Puzzles: Redux
- Ircher
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Ircher He / Him / HisWhat A Grand Idea
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I would think you would have to give a bit more information to help us out on those puzzles; I don't see any patterns.
Not simple addition, subtraction, multiplication, division (try it and you'll see you get different answers for each)
Not modular arithmetic (i.e.: 18 mod X = 5 does not have the same value X as say if one tried 8 mod X = 8 and 24 mod X = 37)
Not non-decimal base system (Some of the sequences increase, others decrease, and in the first one, there is no change and this stuff implies the middle number would be different for each one)
Nothing to do with prime factorization
Doesn't seem to have anything to do with the digit sequence
Also, how can the same input give two different outputs??? (See puzzle B : Apparently, 2 gives both 17 and 31 as possible outputs after going through the middle)- Ircher
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Ircher He / Him / HisWhat A Grand Idea
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In post 138, word321 wrote:1. Prove that % is distributed over addition, that is:
(x+y)%z=(x%z+y%z)%zSpoiler: My Solution
I will note that #2 is much easier to do by directly applying the definitions of even and odd versus using the module operation.Last edited by Ircher on Sat Sep 19, 2020 1:17 pm, edited 2 times in total.- Ircher
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In post 141, StrangerCoug wrote:Let z1 and z2 be two complex numbers whose magnitudes are r1 and r2 respectively and whose phases are φ1 and φ2 respectively. Prove algebraically that z1z2 = r1r2 cis(φ1 + φ2), where cis(φ) = cos(φ) + i sin(φ).Spoiler: My Attempt- Ircher
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Spoiler:- Ircher
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Here is a fun integral to do from my probability and statistics course last semester:
The probability density function for the standard normal distribution is given by $f_Y(y)=ke^{-y^2/2}$ where $k=1 / \sqrt{2\pi}$. Show that the variance of the standard normal distribution is equal to 1 by computing the following integral:
$$\int_{-\infty}^{\infty}{y^2f_Y(y)dy}-\left(\int_{-\infty}^{\infty}{yf_Y(y)dy}\right)^2.$$
(You will probably want to copy the equations inside dollar signs into Desmos or something since the forums don't support rendering math really well at all.) You can show your work by just posting the latex code involved.- Ircher
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Ircher He / Him / HisWhat A Grand Idea
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You should prove the standard normal integral is 1 as well if you are going to use that argument. (Yes, it is a probability density function, so it equals 1, but the solution should be understandable by someone who has only taken math through multivariable calculus. This makes the problem more interesting.) - Ircher
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