Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

How many solutions

How many solutions does the equation

have on the set of positive integers?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: How many solutions

Just use Excel.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,418

Re: How many solutions

0, 1, 99
1, 2, 97
2, 3, 95
the list seems endless!

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: How many solutions

No, the list is finite. Also, I think x, y, z must be positive, not 0.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: How many solutions

All right. When x = 1, y can be any integer from 1 to 98. When x = 2, y can be any integer from 1 to 97. When x = 3, y can be any integer from 1 to 96. And so on, until when x = 98, y can only be 1. Once x and y have been chosen, the value of z is determined. Hence it looks like the total number of solutions is 98 + 97 + … + 2 + 1 = 4851.

Last edited by JaneFairfax (2008-01-01 03:29:15)

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: How many solutions

Reminds me of the base of a pyramid with the origin as the apex and
the center of the base near three points (33,33,34), (34,33,33), and (33,34,33).
This base contains all the points needed I declare warily, I caught half of the flu that others got all of it.

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igloo myrtilles fourmis

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: How many solutions

Here's another way of looking at it:
Think about a number line from 1 to 100. The line has 100 numbers in it, so there are 99 "gaps" between the numbers.

If we place two 'dividers' on the number line, and define x to be the amount left of the dividers, y to be the amount between them and z to be the amount right of them, then tony's question is equivalent to "how many ways can the dividers be placed?"

Clearly, the first divider can be placed in 99 spots and the second can be placed in 98 (99 less the position that the first divider is occupying). Then, the ways to place them are 99x98.

Divide that by two because swapping the two dividers doesn't give a different position, and so the answer is (98x99)/2 = 1+2+...+98.

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Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: How many solutions

I like Mathsy’s solution.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: How many solutions

Cool Neat Awesome!!
Now another way with the tri-pyramid, is to
go on the 3 axes from 0 to 100, which is 101 numbers.
This makes a bowling pin arrangement with 1,2,3,4,5,...101.
Then you strip the axes rows off all three sides which
seems to yield 98 rows, just like Mathsy's answer!!!!  4851 ways.
(I meant strip off the diagonal rows on the outside that connect
between each of the 3 pairs of axes at 100 out on each axis)
So for example, if it were smaller like
vvvv1vvv
vvv2v3vv
vv4v5v6v
v7v8v9vJ, then after stripping off the 1-2-4-7 and the 1-3-6-J and the 7-8-9-J, you
are left with just the 5 in the middle, which is 1 row, or three less than 4 rows you
start with.

Last edited by John E. Franklin (2008-01-03 07:47:44)

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igloo myrtilles fourmis

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: How many solutions

Yes nice answer mathsy! This is a really important method for answering questions like:

A toy store owner sells 10 toys in one hour. If he only manufactures 3 types of toy, how many different ways can he sell his 10 toys?

And more advanced questions, such as question 7 here:

http://www.mathscomp.ms.unimelb.edu.au/ … 2007IS.pdf

Last edited by Identity (2008-01-01 19:50:44)

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: How many solutions

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"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: How many solutions

hi gAr

i have seen you use the term GF for a long time.i tried to search it on the net but it didn't clarify anything.could you explain what that is(maybe in another topic)

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: How many solutions

Hi anonimnystefy,

It's kind of intuitive to me. Can't really explain better than the authors.

Are you sure you searched well?
This is a good introduction found in the first page of search results: www.mathdb.org/notes_download/elementar … ae_A11.pdf

Also, there a complete book written about it, available here: http://www.math.upenn.edu/~wilf/DownldGF.html

Aside, how much did you progress with the limit on the other thread?

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"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: How many solutions

thank for the link.

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: How many solutions

A generating function is a clothesline.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: How many solutions

who is wilf?

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: How many solutions

The author of Generatingfunctionology, the first two chapters are about the best there is. After that it stiffens up.

Also it is freely downloadable on the net.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: How many solutions

Ahhhh... How many stupidities have came from me. The answer looks so simple now...

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.