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Geek Culture / Maths Challenges

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Chris K
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Posted: 17th Jun 2004 01:13
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I wonder if anyone can prove any of these:

1. Fairly easy one to start off with (this is from a GCSE paper) - Prove that, for any integer k, k(k+1)(2k+1) is a multiple of 6

2. Prove that the area of a circle is a quarter of the surface area of a sphere with the same radius. - pretty tricky

3. Prove that the difference between the square of two primes bigger than 3 is a multiple of 24 -

If you can do any then post others or other people.
Post your proofs in a code box so you don't give them away.
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HZence
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Posted: 17th Jun 2004 01:35 Edited at: 17th Jun 2004 01:36
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For the first one...




Team EOD :: Programmer/Storyboard Assistant
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MikeS
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Posted: 17th Jun 2004 02:02 Edited at: 17th Jun 2004 02:04
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Anwser to #2
(May be hard to read because I don't know the fancy symbols.)



Hehe, we just got done using these in math around May. Hope I remembered right.

[EDIT] The anwsers are of course in Square units.

A book? I hate book. Book is stupid.
(Formerly known as Yellow)
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Chris K
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Posted: 17th Jun 2004 02:08
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@ HZence

That's not quite enough. You can't know for sure that all other numbers will work.

I'll give some clues

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andrew11
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Posted: 17th Jun 2004 02:15 Edited at: 17th Jun 2004 02:33
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[ed!t]


"All programmers are playwrites and all computers are lousy actors" -Anon
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HZence
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Posted: 17th Jun 2004 02:17
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Andrew11, you posted basically what I posted. Knotty seems to think it isn't enough


Team EOD :: Programmer/Storyboard Assistant
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andrew11
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Posted: 17th Jun 2004 02:21 Edited at: 17th Jun 2004 02:32
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[edit]


"All programmers are playwrites and all computers are lousy actors" -Anon
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kevil
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Posted: 17th Jun 2004 02:44
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Hmm, I'll try those, I'm used to proving stuff. Let's first do number one.



Now onto the next one

Kevil
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kevil
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Posted: 17th Jun 2004 02:47
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Number 2 confuses me a bit.

I have been taught that the area of a circle is pi*r^2, and that the area of the surface of a sphere is 4*pi*r^2, so then it automatically follows.

Kevil
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kevil
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Posted: 17th Jun 2004 03:33
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For number 3:



Kevil
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kevil
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Posted: 17th Jun 2004 03:48
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Here's an easier one. I actually got this question at an exam:

4. Proof that 23^(1/5) is irrational.

Kevil
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Phaelax
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Posted: 17th Jun 2004 04:40
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I hate proofs! proofs are stupid!

"eureka" - Archimedes
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TheAbomb12
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Posted: 17th Jun 2004 08:22
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Book. I hate book. Book is stupid

Amist the Blue Skies...
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Chris K
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Posted: 17th Jun 2004 11:05
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Whoa. Well done kevil.

Yeah, for the second one, try and prove that the area is four times. Like they would have when they made the formula for SA of a sphere for the first time.

I'll have a go at you're one this evening, I've got a physics exam.
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spooky
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Posted: 17th Jun 2004 12:27
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I did number 2 in A level maths. Trouble is that it was 17 years ago! Had to do come up with all the equations for surface areas and volumes of spheres and areas of circles and the like, from scratch. I think it uses integrals and slicing up the object but my mind's a blank.


Boo!
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kevil
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Posted: 17th Jun 2004 13:38
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We'll, I think there are many ways to calculate the surface area of a sphere. The method that is most recent to me is using surface integrals. Here's how:



Is that enough proof?

Kevil
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Chris K
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Posted: 17th Jun 2004 16:17
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Hmm... I guess it is.
Try it without intergrating. I'm 90% it's possible.
Think of a sphere inside a cylinder that is the same radius.

I think they knew a formula for SA of a sphere before integration was invented.

You can get A = Pi r^2 by integrating with triangles (like you are cutting a cake into slices). The triangles fit together to make a box with height r and length Pi * r (2 pi r /2). The height is r because as the base of isosceles tends to zero, the sides become the same as the height.
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kevil
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Posted: 17th Jun 2004 17:16
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Hmm, might try that later.
I always find it hard to prove something without using certain things you already know. It's annoying.

Anyway, here are a few more challenges for you:

5. Prove that the alternating series sum((-1)^(n-1) / n, for n 1..infinite) converges to ln(2).

So show that:
1/1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ......... = ln(2)

6. Derive a direct formula for the fibonacci series. The fibonacci series is described by:

Un = Un-1 + Un-2, U0 = 0, U1=1

So find Un as a function of only n.

Kevil
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kevil
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Posted: 18th Jun 2004 01:02
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For the sphere in the cilinder:

Do you mean you want to show that the surface of the sphere is the same as the surface of the round side of the cilinder with the same radius (which is 2*pi*r * 2*r = 4*pi*r^2)?

I have no idea how to do that though.

And nobody wants to try my challenges? Weaklings

Kevil
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MikeS
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Posted: 18th Jun 2004 04:22
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Chris, when will the anwsers be posted?



A book? I hate book. Book is stupid.
(Formerly known as Yellow)
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Neofish
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Posted: 19th Jun 2004 00:02
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2. Prove that the area of a circle is a quarter of the surface area of a sphere with the same radius. - pretty tricky



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