Algorithm to find which numbers from a list of size n sum to another number


I have a decimal number (let's call it goal) and an array of other decimal numbers (let's call the array elements) and I need to find all the combinations of numbers from elements which sum to goal.

I have a preference for a solution in C# (.Net 2.0) but may the best algorithm win irrespective.

Your method signature might look something like:

public decimal[][] Solve(decimal goal, decimal[] elements)
9/14/2012 9:39:14 PM

Accepted Answer

Interesting answers. Thank you for the pointers to Wikipedia - whilst interesting - they don't actually solve the problem as stated as I was looking for exact matches - more of an accounting/book balancing problem than a traditional bin-packing / knapsack problem.

I have been following the development of stack overflow with interest and wondered how useful it would be. This problem came up at work and I wondered whether stack overflow could provide a ready-made answer (or a better answer) quicker than I could write it myself. Thanks also for the comments suggesting this be tagged homework - I guess that is reasonably accurate in light of the above.

For those who are interested, here is my solution which uses recursion (naturally) I also changed my mind about the method signature and went for List> rather than decimal[][] as the return type:

public class Solver {

    private List<List<decimal>> mResults;

    public List<List<decimal>> Solve(decimal goal, decimal[] elements) {

        mResults = new List<List<decimal>>();
        RecursiveSolve(goal, 0.0m, 
            new List<decimal>(), new List<decimal>(elements), 0);
        return mResults; 

    private void RecursiveSolve(decimal goal, decimal currentSum, 
        List<decimal> included, List<decimal> notIncluded, int startIndex) {

        for (int index = startIndex; index < notIncluded.Count; index++) {

            decimal nextValue = notIncluded[index];
            if (currentSum + nextValue == goal) {
                List<decimal> newResult = new List<decimal>(included);
            else if (currentSum + nextValue < goal) {
                List<decimal> nextIncluded = new List<decimal>(included);
                List<decimal> nextNotIncluded = new List<decimal>(notIncluded);
                RecursiveSolve(goal, currentSum + nextValue,
                    nextIncluded, nextNotIncluded, startIndex++);

If you want an app to test this works, try this console app code:

class Program {
    static void Main(string[] args) {

        string input;
        decimal goal;
        decimal element;

        do {
            Console.WriteLine("Please enter the goal:");
            input = Console.ReadLine();
        while (!decimal.TryParse(input, out goal));

        Console.WriteLine("Please enter the elements (separated by spaces)");
        input = Console.ReadLine();
        string[] elementsText = input.Split(' ');
        List<decimal> elementsList = new List<decimal>();
        foreach (string elementText in elementsText) {
            if (decimal.TryParse(elementText, out element)) {

        Solver solver = new Solver();
        List<List<decimal>> results = solver.Solve(goal, elementsList.ToArray());
        foreach(List<decimal> result in results) {
            foreach (decimal value in result) {
                Console.Write("{0}\t", value);


I hope this helps someone else get their answer more quickly (whether for homework or otherwise).


5/15/2019 1:44:12 AM

The subset-sum problem, and the slightly more general knapsack problem, are solved with dynamic programming: brute-force enumeration of all combinations is not required. Consult Wikipedia or your favourite algorithms reference.

Although the problems are NP-complete, they are very "easy" NP-complete. The algorithmic complexity in the number of elements is low.

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