Summation Problem: Find Lowest Non-Negative Value

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In summary: I don't really remember where I learned it, but it's a really helpful technique. In summary, the least non-negative value that one can form by adding a + or - in front of each number and summing the values is 100.
  • #1
jwxie
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Homework Statement



A set contains numbers from 1-100. What is the least non-negative value that one can form by putting a + or - in front of each number, and summing the values?

Homework Equations



there are a few general summation formulas which I know...

The Attempt at a Solution



The problem that I am facing is the purpose:

by adding a + or - in front of each number

Is the question asking me to perform an alternating series? +, -, +, -, +, -
or solely one for 1+2+3+4+5... and one for 1-2+3-4+5...

Please give me some guidance... thanks
 
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  • #2
You can add a + or a - anywhere, so you can partition the set and then proceed to + or - accordingly. If the set is the integers in [0,100], then you can pair integers a,b such that a+b=100. Hopefully I am understanding the question right.
 
  • #3
Hmmm let me update the question if anyone is confused.

A set contains numbers from 1-100. What is the least non-negative value that one can form by putting a + or - in front of each number, and summing the values?

I think this is a bit more clear.Hmm I am sorry, VeeEight. I read about partition, but I still don't understand its application.
Now i just rewrite the question, so i will wait for your confirmation.
 
  • #4
My idea was that if the your set is {0, 1, 2, ..., 100}, then you can say 0+100=100, 1+99=100, 2+98=... and so on. Thus, you can take 100/2 terms in this form and take a second class of 100/2 terms and add them up and subtract the two classes.
 
  • #5
We want to consider every such expression. For instance:

1 + 2 + 3 + 4 + ... + 100
where every sign is +.

1 - 2 - 3 + 4 - 5 + 6 - 7 + 8 + 9 + 10 - 11 + ... + 100
where every prime has a - sign.

For each number, we choose a + or a -, and then we evaluate the resulting sum/difference. There's no way to evaluate these in general other than going through each and adding/subtracting all 100 numbers.

However, we're looking for the smallest non-negative sum, so if we can show a way to sum these numbers and get zero, we'd be finished.

Just start experimenting with the signs on the first several numbers, and you should find a nice pattern.

Big hint:
Note that 1 - 2 - 3 + 4 = 5 - 6 - 7 + 8 = 0.
 
  • #6
wooo this is really really impressive. i wish i learn this at my young age. my friend told me he learned this since he was in Math Olympics team.

lol thanks guys.
 

1. What is the "Summation Problem" and why is it important?

The Summation Problem is a mathematical concept that involves finding the lowest non-negative value in a sequence of numbers. It is important because it can be used to solve various real-world problems, such as finding the minimum cost for a set of items or determining the shortest path in a network.

2. How is the lowest non-negative value determined in the Summation Problem?

The lowest non-negative value is determined by adding all the numbers in the sequence and then subtracting the largest negative value. This will give the lowest non-negative value in the sequence.

3. Can the Summation Problem have multiple solutions?

Yes, the Summation Problem can have multiple solutions if there are multiple negative numbers in the sequence. In this case, the lowest non-negative value will be different depending on which negative number is subtracted from the sum.

4. How does the size of the sequence affect the solution to the Summation Problem?

The size of the sequence does not directly affect the solution to the Summation Problem. However, a larger sequence may make it more difficult to determine the lowest non-negative value as there are more numbers to consider.

5. Are there any alternative methods for solving the Summation Problem?

Yes, there are alternative methods for solving the Summation Problem, such as using computer algorithms or mathematical formulas. These methods may be more efficient for larger sequences or more complex problems.

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