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#8926 2013-03-08 20:34:20

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

We sent 8 e-mails to each other yesterday. I'll find out where she lives... maybe we can walk to school together.

Okay, thanks.

#8927 2013-03-08 20:38:16

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,610

Re: Linear Interpolation FP1 Formula

I am coming up with 70 such numbers for 4 digits alone.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#8928 2013-03-08 20:41:24

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Sorry, should have specified, the number has to be 4-digits long, so 70 is the right answer. How did you do it with GFs?

#8929 2013-03-08 20:44:27

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,610

Re: Linear Interpolation FP1 Formula

The answer is the coefficient of x^0, which is 70.

Last edited by bobbym (2013-03-08 20:49:38)


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#8930 2013-03-08 20:54:20

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

I can see that the second bracket represents the 1s, but what is happening in the others?

#8931 2013-03-08 20:55:44

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,610

Re: Linear Interpolation FP1 Formula

The first bracket represents the first digit. It can be from 1 to 4.

The second bracket represents the second digit. It can be from 0 to 4.

Do you follow?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#8932 2013-03-08 20:59:06

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Okay, that makes sense... but I don't understand why you have negative exponents for the last two. Why isn't it just the second bracket cubed since each of the other numbers can be from 0 to 4?

#8933 2013-03-08 21:04:00

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,610

Re: Linear Interpolation FP1 Formula

When you are solving a problem like this you are really solving a linear diophantine equation. Mathematics consists of taking a problem that is in one form and changing into another that has the same number of solutions. One that we can solve. We will get into the "teakettle principle" in just a minute.

The problem really is the number of solutions to

a + b = c + d when 0<a<5, 0 ≤ b,c,d < 5. And a,b,c,d are each a single digit of your 4 digit number.

Follow so far?

Last edited by bobbym (2013-03-08 21:04:32)


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#8934 2013-03-08 21:11:42

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Oh, I see... so it becomes

a + b + c + d = 0

where a and b have the same ranges as before, but now c and d can be between 0 and -4.

which is why we need the co-efficient of x^0, and why we have negative powers...

#8935 2013-03-08 21:17:31

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,610

Re: Linear Interpolation FP1 Formula

That is one way of looking at it. The last two brackets just represent c and d taking on values of -4 to 0.

When we expand that we get:

We see the constant coefficient ( x^0 ) is 70. What do you
think the 34 x^4 tells us?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#8936 2013-03-08 21:21:03

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

The number of solutions to a + b + c + d = 4?

#8937 2013-03-08 21:25:55

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,610

Re: Linear Interpolation FP1 Formula

Yes! It means that there are 34 ways to write that 4 digit number abcd such that a + b is 4 greater than c + d.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#8938 2013-03-08 21:27:58

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Ah, I see. Thanks for the solution! Now on to the second part of the question.

#8939 2013-03-08 21:33:25

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,610

Re: Linear Interpolation FP1 Formula

What second part?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#8940 2013-03-08 21:50:05

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

It asks for how many 4-digit numbers can be made using the numbers 0 to k. I used the same approach, it did not work...

#8941 2013-03-08 22:01:22

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,610

Re: Linear Interpolation FP1 Formula

Hi;

That is more difficult but a tentative experimental answer can be found quickly.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#8942 2013-03-08 22:08:51

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Yes, that's the one -- although, this question is from a public exam paper where only a pen is allowed...

The GF is great for solving the first part, but the second part looks like a systematic approach is needed.

#8943 2013-03-08 22:16:02

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,610

Re: Linear Interpolation FP1 Formula

Yes, but the important part is that we have the answer... Do they agree that is correct?

Then I can work on the so called proof.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#8944 2013-03-08 22:17:33

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Yes. It is okay, I found the 'full solution' -- although, I am trying to see if I can do it with a GF.

#8945 2013-03-08 22:20:51

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,610

Re: Linear Interpolation FP1 Formula

Can you supply the link?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#8946 2013-03-08 22:22:22

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

To the question paper, or the solution?

#8947 2013-03-08 22:25:14

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,610

Re: Linear Interpolation FP1 Formula

To both if you have them. In case I do not get it.

I would try induction before using the gf approach here.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#8948 2013-03-08 22:29:23

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

w w w . m a t h s h e l p e r . c o . u k / o x b . h t m

Under the heading STEP Past Papers, look at Q1 of 2007 Paper 1 for the question, or click on '2007 Solutions' to see the answers.

#8949 2013-03-08 22:29:57

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

How would you use induction?

#8950 2013-03-08 22:34:45

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,610

Re: Linear Interpolation FP1 Formula

Sorry, it did not work. Looking at a gf approach.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

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