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I finally managed to learn a bit of number theory and Diophantine equation(with the help of Arturo Magidin's great answer in what type of math is this?). But I'm wondering what's the next step after?

How do you take all the results and figure out which one is correct? If I use a small sum and very specific numbers then I may only get a handful of results but as my numbers and variables get bigger I get many many results(I stop my computer after a million). I'm new to math to I thought I'd ask the community here what are the approaches to this step now?

I've been thinking and, correct me if I'm wrong, but this is probably a probability question now since every answer is technically correct. I know the correct answer is somewhere in the millions of results I get, but I'm trying to at least narrow it down.

Do I use a computer program to study each result against parameters I set(i.e. no result can change more than 5% and only 2 colors can be chosen at a time), or is there a mathematical way of solving it?

Thanks in advance! (sorry if I'm wording this question incorrectly..I'm not 100% sure the name of what I am searching for)

Here's some more information to clear it up more. For example I have the following:

color  value   quantity red       20    2 blue    5   8 green   10  2  total       100 

If only the value and the total is given, I will find there is 36 possible answers:

#1 Found : 20.0*0.0 red + 5.0*0.0 blue + 10.0*10.0 green = 100.0 #2 Found : 20.0*0.0 red + 5.0*2.0 blue + 10.0*9.0 green = 100.0 #3 Found : 20.0*0.0 red + 5.0*4.0 blue + 10.0*8.0 green = 100.0 #4 Found : 20.0*0.0 red + 5.0*6.0 blue + 10.0*7.0 green = 100.0 #5 Found : 20.0*0.0 red + 5.0*8.0 blue + 10.0*6.0 green = 100.0 #6 Found : 20.0*0.0 red + 5.0*10.0 blue + 10.0*5.0 green = 100.0 #7 Found : 20.0*0.0 red + 5.0*12.0 blue + 10.0*4.0 green = 100.0 #8 Found : 20.0*0.0 red + 5.0*14.0 blue + 10.0*3.0 green = 100.0 #9 Found : 20.0*0.0 red + 5.0*16.0 blue + 10.0*2.0 green = 100.0 #10 Found : 20.0*0.0 red + 5.0*18.0 blue + 10.0*1.0 green = 100.0 #11 Found : 20.0*0.0 red + 5.0*20.0 blue + 10.0*0.0 green = 100.0 #12 Found : 20.0*1.0 red + 5.0*0.0 blue + 10.0*8.0 green = 100.0 #13 Found : 20.0*1.0 red + 5.0*2.0 blue + 10.0*7.0 green = 100.0 #14 Found : 20.0*1.0 red + 5.0*4.0 blue + 10.0*6.0 green = 100.0 #15 Found : 20.0*1.0 red + 5.0*6.0 blue + 10.0*5.0 green = 100.0 #16 Found : 20.0*1.0 red + 5.0*8.0 blue + 10.0*4.0 green = 100.0 #17 Found : 20.0*1.0 red + 5.0*10.0 blue + 10.0*3.0 green = 100.0 #18 Found : 20.0*1.0 red + 5.0*12.0 blue + 10.0*2.0 green = 100.0 #19 Found : 20.0*1.0 red + 5.0*14.0 blue + 10.0*1.0 green = 100.0 #20 Found : 20.0*1.0 red + 5.0*16.0 blue + 10.0*0.0 green = 100.0 #21 Found : 20.0*2.0 red + 5.0*0.0 blue + 10.0*6.0 green = 100.0 #22 Found : 20.0*2.0 red + 5.0*2.0 blue + 10.0*5.0 green = 100.0 #23 Found : 20.0*2.0 red + 5.0*4.0 blue + 10.0*4.0 green = 100.0 #24 Found : 20.0*2.0 red + 5.0*6.0 blue + 10.0*3.0 green = 100.0 #25 Found : 20.0*2.0 red + 5.0*8.0 blue + 10.0*2.0 green = 100.0 #26 Found : 20.0*2.0 red + 5.0*10.0 blue + 10.0*1.0 green = 100.0 #27 Found : 20.0*2.0 red + 5.0*12.0 blue + 10.0*0.0 green = 100.0 #28 Found : 20.0*3.0 red + 5.0*0.0 blue + 10.0*4.0 green = 100.0 #29 Found : 20.0*3.0 red + 5.0*2.0 blue + 10.0*3.0 green = 100.0 #30 Found : 20.0*3.0 red + 5.0*4.0 blue + 10.0*2.0 green = 100.0 #31 Found : 20.0*3.0 red + 5.0*6.0 blue + 10.0*1.0 green = 100.0 #32 Found : 20.0*3.0 red + 5.0*8.0 blue + 10.0*0.0 green = 100.0 #33 Found : 20.0*4.0 red + 5.0*0.0 blue + 10.0*2.0 green = 100.0 #34 Found : 20.0*4.0 red + 5.0*2.0 blue + 10.0*1.0 green = 100.0 #35 Found : 20.0*4.0 red + 5.0*4.0 blue + 10.0*0.0 green = 100.0 #36 Found : 20.0*5.0 red + 5.0*0.0 blue + 10.0*0.0 green = 100.0 

As you can see, in the possibilities I get the correct answer but many other answers also. Now say I add one more red(so the total red is 3) then I now have 49 results, but some of the results in second set are not likely if you factor in the relationship with the first result set. I assume as I get more data results, I can more accurately remove the results that don't work. I'm not sure if there's a branch of math that deals with this?

Update: Is this probability? fitness(or cost) functions? mathematical optimization? or is this even a math problem?

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    Been searching and is what I am looking for called a fitness function?2011-06-07

1 Answers 1

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I can't understand what you want, so I'll post a simple example and maybe you can clarify. Suppose reds are worth 7, blues are worth 12, and the total value is to be 67. Then there's only one solution: 1 red, and 5 blue. Now change that 67 to 68, and there's still only one solution, and it's 8 red, and 1 blue. So both the number of reds and the number of blues have changed dramatically, which seems to be something you wish to avoid - but you can't avoid it, it's inescapable for this problem.

My guess is that in any setting one can come up with some numbers where small changes in total value force large changes in quantities.

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    Its hard to narrow it down when all answers are so correct..This is why I thought there might be some frameworks or best practices to help narrow the choices down.2011-06-08