Question

Supposing I’m having the constants 3,5,6,9,10. How can I detect how to write $n, which is the input, as a sum of these constants with the least number of terms?

Examples

$n=10, S=10
$n=18, S=9+9
$n=24, S=9+9+6
$n=27, S=9+9+9
$n=28, S=10+9+9

Thanks

Answer 1

This is another Python solution, but hopefully it’s easy for you to convert to PHP (I would do it myself, but I’m no PHP expert – I’m sure you could do a better job of it). I’ve tried not to use any advanced Python funcitons, so that it is easier for non-Python readers to understand, but if some Python syntax is not clear, please just ask.

allowed = [3, 5, 6, 9, 10]
n = 28

solutions = [ None ] * (n + 1)
solutions[0] = []

for i in range(n + 1):
    if solutions[i] is None: continue
    for a in allowed:
        if i + a > n: continue
        if solutions[i + a] is None or len(solutions[i]) + 1 < len(solutions[i + a]):
            solutions[i + a] = solutions[i] + [a]

print solutions[28]

It works by starting from 0 and building up to the desired number, keeping a cache of the shortest solution seen so far for each possible total. It has a running time of O(n * a), where a is the number of different allowed values.

By the way, your answer to n=28 is wrong. It should be [9, 9, 10].

Update: here’s my attempt at a PHP solution:

<?php
$allowed = array(3, 5, 6, 9, 10);
$n = 28;

$solutions = array();
$solutions[0] = array();

foreach (range(0, $n) as $i) {
    if (is_null($solutions[$i])) continue;
    foreach ($allowed as $a) {
        if ($i + $a > $n) continue;
        if (is_null($solutions[$i + $a]) ||
            sizeof($solutions[$i]) + 1 < sizeof($solutions[$i + $a])) {
            $solutions[$i + $a] = array_merge($solutions[$i], array($a));
        }
    }
}

var_dump($solutions[$n]);
?>

It gives the right answer, but please be aware that I’m not a professional PHP coder – I just looked up the equivalent functions in the PHP documentation.