1) Start with the FOUR final states.
2) Use only 15 states.
3) If it doesn't work out, then read the hint further.
unsigned char constexpr stopStates = 4;
auto constexpr go = []() {//state machine (2D array)
    std::array<std::array<decltype(+stopStates),256>,15> g;
    for(auto&e : g)    //default
        std::fill(e.begin(), e.end(), g.size());//wrong state
    for(size_t i = 0; i!=stopStates; ++i)    //default
        g[i]['i'] = stopStates, g[i]['o'] = stopStates+1;
    //===== stop states =====
    g[0]['p'] = 8;   //puton
    //in \ out
    g[1]['p'] = 12;  //put
    //input \ output
    g[2]['o'] = 14;  //on
    g[2]['p'] = 8;   //puton
    //put_on
    g[3]['p'] = 8;   //puton
    g[3]['e'] = 0;   //e
    //===== non stop states =====
    g[4]['n'] = 1;   // i -> in
    g[5]['n'] = 6;   // o -> on
    g[5]['u'] = 7;   // o -> ou
    g[6]['e'] = 0;   // on -> one
    g[7]['t'] = 1;   // ou -> out
    g[8]['u'] = 9;   // p -> pu
    g[9]['t'] = 10;  // pu -> put
    g[10]['o'] = 11; // put -> puto
    g[11]['n'] = 0;  // puto -> puton
    g[12]['u'] = 13; // ?p -> ?pu
    g[13]['t'] = 2;  // ?pu -> ?put
    g[14]['n'] = 3;  // ?puto -> ?puton
    g[14]['u'] = 7;  // ?puto -> ?putou
    return g;
    //todo: Timus - automatic construction of a MINIMAL state machine (NP ?)
}();