This is a classic interview question from Microsoft research Asia.
How to Find 2 numbers in an unsorted array equal to a given sum.

[1]brute force solution
This algorithm is very simple. The time complexity is O(N^2)

[2]Using binary search
Using bianry searching to find the Sum-arr[i] with every arr[i], The time complexity can be reduced to O(N*logN)

[3]Using Hash
Base on [2] algorithm and use hash, the time complexity can be reduced to O(N), but this solution will add the O(N) space of hash.

[4]Optimal algorithm:

Pseduo-code:

for(i=0;j=n-1;i<j)
if(arr[i]+arr[j]==sum) return (i,j);
else if(arr[i]+arr[j]<sum) i++;
else j--;
return(-1,-1);

or

If a[M] + a[m] > I then M--
If a[M] + a[m] < I then m++
If a[M] + a[m] == I you have found it
If m > M, no such numbers exist.

And, Is this quesiton completely solved? No. If the number is N. This problem will become very complex.

The quesiton then:
How can I find all the combination cases with a given number?

This is a classic NP-Complete problem which is called subset-sum.
To understand NP/NPC/NP-Hard you'd better to read some professional books.

References:
[1]http://www.quora.com/Mathematics/How-can-I-find-all-the-combination-cases-with-a-given-number
[2]http://en.wikipedia.org/wiki/Subset_sum_problem

This is a classic interview question from Microsoft research Asia.
How to Find 2 numbers in an unsorted array equal to a given sum.

[1]brute force solution
This algorithm is very simple. The time complexity is O(N^2)

[2]Using binary search
Using bianry searching to find the Sum-arr[i] with every arr[i], The time complexity can be reduced to O(N*logN)

[3]Using Hash
Base on [2] algorithm and use hash, the time complexity can be reduced to O(N), but this solution will add the O(N) space of hash.

[4]Optimal algorithm:

Pseduo-code:

for(i=0;j=n-1;i<j) if(arr[i]+arr[j]==sum) return (i,j); else if(arr[i]+arr[j]<sum) i++; else j--; return(-1,-1); 

or

If a[M] + a[m] > I then M-- If a[M] + a[m] < I then m++ If a[M] + a[m] == I you have found it If m > M, no such numbers exist. 

And, Is this quesiton completely solved? No. If the number is N. This problem will become very complex.

The quesiton then:
How can I find all the combination cases with a given number?

This is a classic NP-Complete problem which is called subset-sum.
To understand NP/NPC/NP-Hard you'd better to read some professional books.

References:
[1]http://www.quora.com/Mathematics/How-can-I-find-all-the-combination-cases-with-a-given-number
[2]http://en.wikipedia.org/wiki/Subset_sum_problem