Question

For example an input of 3 would return [1,1,1], [2,1], and [1,2]

I know a lot of combination/permutation problems involve a recursive function that calls itself inside of a loop but I’m unable to see an appropriate way to apply it to this problem.

It’s a concept I’m trying to grasp, here’s what I have so far…

function numberToAddends($number, $arr, $k){
    for ($i = 0; $i < $number; $i++) {
        $arr[$k] = $i;
        numberToAddends($k-$i, $arr, $k + 1);
    }

    if($k <=0){
        print_r($arr);
    }
}

For test input, you can use something like numberToAddends(3, $arr, 0);

Am I thinking along the right path? Can anyone provide complete php syntax to solve this problem along with commented code?

Answer 1

Here’s an algorithm that solves this problem by reusing the sub-result from 1 to n:

$number = 3;
$addends = array();
for ($x=1; $x<=$number; $x++) {
    $addends[$x] = array();
    for ($y=$x-1; $y>0; $y--) {
        foreach ($addends[$y] as $z) {
            $addends[$x][] = array_merge($z, array($x-$y));
        }
    }
    $addends[$x][] = array($x);
}
var_dump($addends);

This builds the result set for x by combining any already known result z for any y < x with the difference of x and y and finally x itself.