count together numbers in a string until only one number from 1 - 9 remains [??vortex math??]

reduce.sh

#!/bin/bash

STRING=$1
NUM=0
until [[ $STRING -lt 10 ]] do;
        foreach n in $(echo $STRING | fold -w1); do
                echo "$NUM + $n";
                ((NUM=NUM+n));
                echo " = $NUM";
        done;
        ((STRING=NUM));
        ((NUM=0));
done;