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I am going to appear at this olympiad:

Link

Since I am only in the 9th grade,

Would someone please suggest a way to improve my Mathematics?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi Agnishom;

The olympiad questions are difficult. To increase you ability at solving problems there is only one way really. That is by doing problems.

I would like to see you participating more in the problems that are posed here. They will help turn you into a problem solver.

Since I am only in the 9th grade,

You are old enough for more.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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Any good book that I can read?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

You were asking about integer partitions on another forum. Do you know anything about generating functions?

Good books are very hard to find. Books that instruct are even harder to find. You can get good one on one instruction right here free.

Have you been continuing your work on that programming site?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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Have you been continuing your work on that programming site?

I am not sure which website you are talking about

Do you know anything about generating functions?

I am confused about that. I can only understand the recursive solution

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Remember, Project Euler?

I am confused about that. I can only understand the recursive solution

First thing I would suggest is the same thing I do when conversing with anonimnystefy, I urge you to abandon the theoretical portion of math that they cram down your throat in school in favor of concrete problems.

Let's not talk about generating functions as a whole. Forget about the properties and fabulous theorems that theoretical mathematicians love. No one ever hires those guys, they hire problem solvers.

I asked on the other forum if you would post the problem. The real life problem which asks how may ways can you arrange this or that.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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Yes, Project Euler!

Forget about the properties and fabulous theorems that theoretical mathematicians love. No one ever hires those guys, they hire problem solvers.

Why?

I want to learn Mathematics to gain knowledge and because I like it not because I want to get hired

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi;

Would you rather know a lot of mathematics or be able to do a lot of mathematics?

Which is more important? Knowing every single proof by Euler or being able to solve problems that Euler solved? People think one thing leads to another but that is not true.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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I understand now

People think one thing leads to another but that is not true.

Therefore, you do not believe in chain reactions?

Anyway, please explain generating functions

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Therefore, you do not believe in chain reactions?

I believe in seeds. A small seed can grow into a large tree. Poison on the other hand does not promote growth. It just kills you.

If you want to compete in olympiads many of the problems will be combinatoric in nature. In combinatorics we count without counting.

Generating functions to make a point are never taught. What is taught are primitive methods of counting that rely on nCr's and nPr's. These are difficult for students to use and understand and therefore they do poorly. They end up hating mathematics and going into other fields.

Simply stated a generating function is a polynomial whose coefficients are integers and are the answers to many of the toughest combinatoric problems.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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Generating functions to make a point are never taught. What is taught are primitive methods of counting that rely on nCr's and nPr's. These are difficult for students to use and understand and therefore they do poorly. They end up hating mathematics and going into other fields.

??

nCr's and nPr's are not that difficult!

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

When you have to do a tough combinatorics problem then they can be cumbersome. They rely on casework to get the answer which is really like counting which is what you are trying to avoid in combinatorics to begin with.

GF's on the other hand are just polynomial arithmetic which everyone understands and knows how to do.

I am not sure which website you are talking about

This is the one:

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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The Problem was:

Let p(n) be the number of ways in which n number of coins can be split into piles.

Now, What is the minimum value of n which p(n) is divisible by one million.

I wish mathisfun.com had a page on GFs

*Last edited by Agnishom (2012-09-14 02:30:43)*

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi Agnishom;

That problem is an unrestricted partition problem. You should start with Euler's pentagonal formula. But it will be difficult.

I wish mathisfun.com had a page on GFs

It may soon have one.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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Please explain this one; I am not aware of the big Pi notation

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

That means a product instead of a series.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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The series is infinite!!

But p(n) should be finite

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

That is a recurrence relation. If you want p(8) you you do not use anything past 8. In this case you would stop at p(7).

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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Where is the number above the Sigma??

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Sometimes, they mean it goes to infinity and they leave it out. Sometimes it goes to n and they leave it out. It really depends on the context.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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So this is recursion; So would I get the error?

`Maximum Recursion Depth reached`

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

That is not exactly the same thing. In some languages that happens when you define a variable in terms of itself.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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The identity implies a marvelous recurrence for calculating p(n), the number of partitions of n:

or more formally,

where the summation is over all nonzero integers k (positive and negative) and g_k is the kth pentagonal number.

But why p(n-7)??

7 isnt a pentagonal number

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi;

You use the generalized pentagonal numbers. They are:

0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100,...

They are not the same as the pentagonal numbers.

The notation is really not doing anything but to obscure the beauty of Euler's idea. An example is worth 10000 lines of notation. Want to see one?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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Ya! ^_^

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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