how to find all possible solutions to a formula, like 100*7-8*3+7? (8 Out of 10 Cats Does Countdown solver)

Question

so as fun i decided to write a simple program that can solve the 8 Out of 10 Cats Does Countdown number puzzle, link is form Countdown, but same rules. so my program simply goes through a all possible combinations of AxBxCxDxExF, where letters are numbers and "x" are +, -, / and *. here is the code for it:

private void combineRecursive( int step, int[] numbers, int[] operations, int combination[]){
    if( step%2==0){//even steps are numbers
        for( int i=0; i<numbers.length; i++){
            combination[ step] = numbers[ i];
            if(step==10){//last step, all 6 numbers and 5 operations are placed
                int index = Solution.isSolutionCorrect( combination, targetSolution);
                if( index>=0){
                    solutionQueue.addLast( new Solution( combination, index));
                }
                return;
            }
            combineRecursive( step+1, removeIndex( i, numbers), operations, combination);
        }
    }else{//odd steps are operations
        for( int i=0; i<operations.length; i++){
            combination[ step] = operations[ i];
            combineRecursive( step+1, numbers, operations, combination);
        }
    }
}

and here is what i use to test if the combination is what i want to not.

public static int isSolutionCorrect( int[] combination, int targetSolution){
    double result = combination[0];
    //just a test
    if( Arrays.equals( combination, new int[]{100,'*',7,'-',8,'*',3,'+',7,'+',50})){
        System.out.println( "found");
    }
    for( int i=1; i<combination.length; i++){
        if(i%2!=0){
            switch( (char)combination[i]){
                case '+': result+= combination[++i];break;
                case '-': result-= combination[++i];break;
                case '*': result*= combination[++i];break;
                case '/': result/= combination[++i];break;
            }
        }
        if( targetSolution==result){
            return i;
        }
    }       
    return targetSolution==result?0:-1;
}

so in last episode i found a problem with my code. this was the solution to one of the puzzles.

(10*7)-(8*(3+7))

i noticed that i do find this combination "10*7-8*3+7" (twice), but because i check for solutions by doing the operation left to right i actually don't find all answers. i only check for solutions like this ((((10*7)-8)*3)+7). so even though i found the combination i don't have the right order.

so now the question is how do i test all possible math orders, like (10*7)-(8*(3+7)), (10*7)-((8*3)+7) or 10*(7-8)*(3+7)? i though i can use a balance tree with operations as balancing nodes. but still i have not idea how to go through all possible combinations without moving around the formula.

(10*7)-(8*(3+7))
          -
     /        \
    *         *
  /   \      /  \
700   7     8    +
                / \
              7    3

(10*7)-((8*3)+7)
          -
     /        \
    *         +
  /   \      /  \
700   7     *    7
           / \
          8  3

10*(7-8)*(3+7)

                 *
           /           \
          *
     /        \         10
    -          +
  /   \      /  \
7     8     3    7

how do i do this in code? not looking for solved code more of how i should change perspective to fix it. i don't know why i am stumped at it.

about me: 4th year computer science, not new or noob at programming (i like to believe at least ;))

Answer 1

This is easier solved with a dedicated class that represents an expression, and not with an array. Then you can simply enumerate all possible trees. A mix of another answer that I wrote for a similar task, and an answer that shows how to generate all binary trees gave this:

import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;

public class NumberPuzzleWithCats
{
    public static void main(String[] args)
    {
        List<Integer> numbers = Arrays.asList(10,7,8,3,7);
        solve(numbers);
    }

    private static void solve(List<Integer> numbers)
    {
        List<Node> nodes = new ArrayList<Node>();
        for (int i=0; i<numbers.size(); i++)
        {
            Integer number = numbers.get(i);
            nodes.add(new Node(number));
        }
        System.out.println(nodes);
        List<Node> all = create(nodes);
        System.out.println("Found "+all.size()+" combinations");


        for (Node node : all)
        {
            String s = node.toString();
            System.out.print(s);
            if (s.equals("((10*7)-(8*(3+7)))"))
            {
                System.out.println(" <--- There is is :)");
            }
            else
            {
                System.out.println();
            }
        }
    }

    private static List<Node> create(Node n0, Node n1)
    {
        List<Node> nodes = new ArrayList<Node>();
        nodes.add(new Node(n0, '+', n1));
        nodes.add(new Node(n0, '*', n1));
        nodes.add(new Node(n0, '-', n1));
        nodes.add(new Node(n0, '/', n1));
        nodes.add(new Node(n1, '+', n0));
        nodes.add(new Node(n1, '*', n0));
        nodes.add(new Node(n1, '-', n0));
        nodes.add(new Node(n1, '/', n0));
        return nodes;
    }

    private static List<Node> create(List<Node> nodes)
    {
        if (nodes.size() == 1)
        {
            return nodes;
        }
        if (nodes.size() == 2)
        {
            Node n0 = nodes.get(0);
            Node n1 = nodes.get(1);
            return create(n0, n1);
        }
        List<Node> nextNodes = new ArrayList<Node>();
        for (int i=1; i<nodes.size()-1; i++)
        {
            List<Node> s0 = create(nodes.subList(0, i));
            List<Node> s1 = create(nodes.subList(i, nodes.size()));
            for (Node n0 : s0)
            {
                for (Node n1 : s1)
                {
                    nextNodes.addAll(create(n0, n1));
                }
            }
        }
        return nextNodes;
    }

    static class Node
    {
        int value;
        Node left;
        Character op;
        Node right;

        Node(Node left, Character op, Node right)
        {
            this.left = left;
            this.op = op;
            this.right = right;
        }
        Node(Integer value)
        {
            this.value = value;
        }

        @Override
        public String toString()
        {
            if (op == null)
            {
                return String.valueOf(value);
            }
            return "("+left.toString()+op+right.toString()+")";
        }
    }
}

It will print all combinations that are created, including the one that you have been looking for:

[10, 7, 8, 3, 7]
Found 16384 combinations
(10+(7+(8+(3+7))))
(10*(7+(8+(3+7))))
...
((10*7)+(8*(3+7)))
((10*7)*(8*(3+7)))
((10*7)-(8*(3+7))) <--- There is is :)
((10*7)/(8*(3+7)))
((8*(3+7))+(10*7))
...
((7/3)-((8/7)/10))
((7/3)/((8/7)/10))

Of course, checking whether the right solution is found by comparing the String representations is ... "very pragmatic", to state it that way, but I think the actual approach of the generation is what was important here.

(I hope this really is what you have been looking for - I could not view the site that you linked to...)