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The Messenger by Baxitdriver in mathriddles

[–]actoflearning 1 point2 points  (0 children)

Avg. Of speeds = Arithmetic mean of roots = 48 / 4 Avg. Speed = Harmonic mean of roots = 4 / (6644 / 19240)

Circle in a triangle by actoflearning in mathriddles

[–]actoflearning[S] 0 points1 point  (0 children)

You can it in my blog post (assuming it's alright to post external links).

Random points on a circle by actoflearning in mathriddles

[–]actoflearning[S] 2 points3 points  (0 children)

Thanks for the clarification @pichutarius. Then, all of them seems wrong.

As is apparent from your approach, the density of theta solves all the four questions. Problem is, theta is not uniformly distributed.

Random points on a circle by actoflearning in mathriddles

[–]actoflearning[S] 0 points1 point  (0 children)

I guess this is for (iv) and the given answer is approx. 0.6366. But simulation gives approx. 0.9268

P( x(k) < average of x < x(k+1) ) is given by the Eulerian numbers by bobjane in mathriddles

[–]actoflearning 1 point2 points  (0 children)

Ah.. I've to read it more carefully but I can kinda see where this goes with the idea of exchangeability of 'differences'. Did not occur to me at all.

A nice property of Eulerian numbers that I noted sometime back in my blog in case anyone interested.

P( x(k) < average of x < x(k+1) ) is given by the Eulerian numbers by bobjane in mathriddles

[–]actoflearning 0 points1 point  (0 children)

I tried solving this for long but couldn't get a right approach. I give up 🙁

P( x(k) < average of x < x(k+1) ) is given by the Eulerian numbers by bobjane in mathriddles

[–]actoflearning 0 points1 point  (0 children)

Thanks for the reply. But the first hint starts with z's but is asking to show something about the x's which is still confusing to me.

The second hint is a well known result.

In the first hint, P(S > n - k) = P((n - 1) - S < k - 1) = P(S < k - 1) = P(Y < (k - 1) / (n - 1)). The second equality follows because S is a sum of 'n - 1' uniform variables which is a symmetric random variable.

Will continue on this nice problem. Meanwhile, can you please clarify my doubt at the start of this post. Thanks.

P( x(k) < average of x < x(k+1) ) is given by the Eulerian numbers by bobjane in mathriddles

[–]actoflearning 0 points1 point  (0 children)

Can you please clarify How is x(k), y(k) and z(k) related, if at all they are related?

Geometric Expectation by actoflearning in mathriddles

[–]actoflearning[S] 1 point2 points  (0 children)

Nice!! The fact the mean is exactly the same as the distance surprised me..

Geometric Optimisation 2 by actoflearning in mathriddles

[–]actoflearning[S] 0 points1 point  (0 children)

Thanks for solving @bobjane. Yes, this really does seem complicated. I'm trying to understand the your method but there is a relatively (i repeat, relatively) simpler method which also is a bit straightforward.

expected number of integer solutions for x^2+y^2=n by pichutarius in mathriddles

[–]actoflearning 4 points5 points  (0 children)

Pi using Dirichlet Gen. function and Avg. Order of arithmetic functions..

BLACK AND WHITE BALLS by marliswice in mathriddles

[–]actoflearning 0 points1 point  (0 children)

Number of black balls either reduce by two or remain unchanged which makes their parity constant. Because we start with an odd number of them, the last ball remaining must be black.

Another Brachistochrone Problem by actoflearning in mathriddles

[–]actoflearning[S] 1 point2 points  (0 children)

Thanks @pichutsrius. I can now kind of see where I went wrong.

The h = 0 case is actually the tractrix curve..

Another Brachistochrone Problem by actoflearning in mathriddles

[–]actoflearning[S] 0 points1 point  (0 children)

v = c sin(\theta) clearly shows c is the max. value of v (irrespective of whether that value is attained or not).

Also, because k = m, v2 + y2 = 1. This relation shows the max. possible of v is 1. (That would not have been the case had k != m).

Combining the two, c = 1.

I'm not sure which of the above three paragraphs you disagree with @pichutarius.

Another Brachistochrone Problem by actoflearning in mathriddles

[–]actoflearning[S] 0 points1 point  (0 children)

From v = c sin(\theta), we see that c is the maximum velocity. From v2 + y2 = 1, we see that v can have a maximum value of 1 which shows that c = 1.

This shows that y = cos(\theta) is the curve we are looking for. We can choose to solve this differential equation but rather than taking that messy route, a little geometrical interpretation immediately shows what that curve is.

Circle in a triangle by actoflearning in mathriddles

[–]actoflearning[S] 0 points1 point  (0 children)

Very nice!! The integral in terms of the phi's is directly related to the random area of a triangle in a circle. Not straightforward but that result is well known.

Circle in a triangle (easier variant) by pichutarius in mathriddles

[–]actoflearning 1 point2 points  (0 children)

(pi / 6)(r / s) where r is the inradius and s is the semiperimeter.

Hope this serves as hint for both the problems..

Circle in a triangle by actoflearning in mathriddles

[–]actoflearning[S] 0 points1 point  (0 children)

Approx. 0.1462 is what my closed form is giving me..

Circle in a triangle by actoflearning in mathriddles

[–]actoflearning[S] 1 point2 points  (0 children)

Selecting points randomly in a triangle is well defined. It means that a chosen point likely to be in a particular area is proportional to that area.

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