void hanoi(vector<VI>& moves, int n_twrs, int n_dsks, VI& aux) {
    int start = aux[0];
    int goal = aux[1];

    if (n_dsks == 1)
        moves.push_back(VI { aux[0], aux[1] });

    else {
        int k = k_hanoi[n_twrs][n_dsks];

        // moves top k disks from start to sparing tower
        int sparing = aux[n_twrs-1];
        aux[1] = sparing;
        aux[n_twrs-1] = goal;
        hanoi(moves, n_twrs, k, aux);

        // moves bottom n-k disks from start to goal
        aux[0] = start;
        aux[1] = goal;
        hanoi(moves, n_twrs - 1, n_dsks - k, aux);

        // moves original k disks from sparing tower to goal
        aux[0] = sparing;
        aux[n_twrs-1] = start;
        hanoi(moves, n_twrs, k, aux);
    }
}