/* #region Head */ // #define _GLIBCXX_DEBUG #include using namespace std; using ll = long long; using ull = unsigned long long; using ld = long double; using pll = pair; template using vc = vector; template using vvc = vc>; using vll = vc; using vvll = vvc; using vld = vc; using vvld = vvc; using vs = vc; using vvs = vvc; template using um = unordered_map; template using pq = priority_queue; template using pqa = priority_queue, greater>; template using us = unordered_set; #define REP(i, m, n) for (ll i = (m), i##_len = (ll)(n); i < i##_len; ++(i)) #define REPM(i, m, n) for (ll i = (m), i##_max = (ll)(n); i <= i##_max; ++(i)) #define REPR(i, m, n) for (ll i = (m), i##_min = (ll)(n); i >= i##_min; --(i)) #define REPD(i, m, n, d) for (ll i = (m), i##_len = (ll)(n); i < i##_len; i += (d)) #define REPMD(i, m, n, d) for (ll i = (m), i##_max = (ll)(n); i <= i##_max; i += (d)) #define REPI(itr, ds) for (auto itr = ds.begin(); itr != ds.end(); itr++) #define ALL(x) begin(x), end(x) #define SIZE(x) ((ll)(x).size()) #define PERM(c) \ sort(ALL(c)); \ for (bool c##p = 1; c##p; c##p = next_permutation(ALL(c))) #define UNIQ(v) v.erase(unique(ALL(v)), v.end()); #define CEIL(a, b) (((a) + (b)-1) / (b)) #define endl '\n' #define sqrt sqrtl #define floor floorl #define log2 log2l constexpr ll INF = 1'010'000'000'000'000'017LL; constexpr int IINF = 1'000'000'007LL; constexpr ll MOD = 1'000'000'007LL; // 1e9 + 7 // constexpr ll MOD = 998244353; constexpr ld EPS = 1e-12; constexpr ld PI = 3.14159265358979323846; template istream &operator>>(istream &is, vc &vec) { // vector 入力 for (T &x : vec) is >> x; return is; } template ostream &operator<<(ostream &os, vc &vec) { // vector 出力 (for dump) os << "{"; REP(i, 0, SIZE(vec)) os << vec[i] << (i == i_len - 1 ? "" : ", "); os << "}"; return os; } template ostream &operator>>(ostream &os, vc &vec) { // vector 出力 (inline) REP(i, 0, SIZE(vec)) os << vec[i] << (i == i_len - 1 ? "\n" : " "); return os; } template istream &operator>>(istream &is, pair &pair_var) { // pair 入力 is >> pair_var.first >> pair_var.second; return is; } template ostream &operator<<(ostream &os, pair &pair_var) { // pair 出力 os << "(" << pair_var.first << ", " << pair_var.second << ")"; return os; } // map, um, set, us 出力 template ostream &out_iter(ostream &os, T &map_var) { os << "{"; REPI(itr, map_var) { os << *itr; auto itrcp = itr; if (++itrcp != map_var.end()) os << ", "; } return os << "}"; } template ostream &operator<<(ostream &os, map &map_var) { return out_iter(os, map_var); } template ostream &operator<<(ostream &os, um &map_var) { os << "{"; REPI(itr, map_var) { auto [key, value] = *itr; os << "(" << key << ", " << value << ")"; auto itrcp = itr; if (++itrcp != map_var.end()) os << ", "; } os << "}"; return os; } template ostream &operator<<(ostream &os, set &set_var) { return out_iter(os, set_var); } template ostream &operator<<(ostream &os, us &set_var) { return out_iter(os, set_var); } template ostream &operator<<(ostream &os, pq &pq_var) { pq pq_cp(pq_var); os << "{"; if (!pq_cp.empty()) { os << pq_cp.top(), pq_cp.pop(); while (!pq_cp.empty()) os << ", " << pq_cp.top(), pq_cp.pop(); } return os << "}"; } void pprint() { cout << endl; } template void pprint(Head &&head, Tail &&... tail) { cout << head; if (sizeof...(Tail) > 0) cout << ' '; pprint(move(tail)...); } // dump #define DUMPOUT cerr void dump_func() { DUMPOUT << endl; } template void dump_func(Head &&head, Tail &&... tail) { DUMPOUT << head; if (sizeof...(Tail) > 0) DUMPOUT << ", "; dump_func(move(tail)...); } // chmax (更新「される」かもしれない値が前) template > bool chmax(T &xmax, const U &x, Comp comp = {}) { if (comp(xmax, x)) { xmax = x; return true; } return false; } // chmin (更新「される」かもしれない値が前) template > bool chmin(T &xmin, const U &x, Comp comp = {}) { if (comp(x, xmin)) { xmin = x; return true; } return false; } // ローカル用 #define DEBUG_ #ifdef DEBUG_ #define DEB #define dump(...) \ DUMPOUT << " " << string(#__VA_ARGS__) << ": " \ << "[" << to_string(__LINE__) << ":" << __FUNCTION__ << "]" << endl \ << " ", \ dump_func(__VA_ARGS__) #else #define DEB if (false) #define dump(...) #endif #define VAR(type, ...) \ type __VA_ARGS__; \ cin >> __VA_ARGS__; template istream &operator,(istream &is, T &rhs) { return is >> rhs; } template ostream &operator,(ostream &os, const T &rhs) { return os << ' ' << rhs; } struct AtCoderInitialize { static constexpr int IOS_PREC = 15; static constexpr bool AUTOFLUSH = false; AtCoderInitialize() { ios_base::sync_with_stdio(false), cin.tie(nullptr), cout.tie(nullptr); cout << fixed << setprecision(IOS_PREC); if (AUTOFLUSH) cout << unitbuf; } } ATCODER_INITIALIZE; void Yn(bool p) { cout << (p ? "Yes" : "No") << endl; } void YN(bool p) { cout << (p ? "YES" : "NO") << endl; } /* #endregion */ // #include // using namespace atcoder; // 先手は,箱を先頭からいくつか開封済みにする. // 全部開封済みにして Grundy 数 0 を相手の手番に渡せるならよいが,そうでないなら // 先手は末尾側のいくつかの箱を未開封で残しておくしか手がない // Problem void solve() { VAR(ll, n); vll a(n); cin >> a; /* i 箱開封された状態で,開封済みの箱の Grundy 数を b[0], b[1], ..., b[i-1] とする. このときの未開封のすべての箱 (i, i+1, ..., n-1) 部分の Grundy 数を u[i] とおくと, i 箱開封された状態全体の Grundy 数は b[0] ^ b[1] ^ ... ^ b[i-1] ^ u[i] である. ここで,u[i] 部分,すなわち未開封のすべての箱 (i, i+1, ..., n-1) 部分のみを考えると,こ こから遷移可能な状態は以下である. - a[i] ^ u[i+1] (1 箱だけ開封した) - a[i] ^ a[i+1] ^ u[i+2] (2 箱だけ開封した) - a[i] ^ a[i+1] ^ a[i+2] ^ u[i+3](3 箱だけ開封した) - ... - a[i] ^ a[i+1] ^ a[i+2] ^ ... ^ a[n-1] ^ u[n](n 箱すべて開封した) - ここで, u[n] = 0 である(n 箱開封された状態で,開封済みの箱に関しては操作不能なので). よって,以下がいえる. u[i] = mex{ a[i] ^ u[i+1], a[i] ^ a[i+1] ^ u[i+2], a[i] ^ a[i+1] ^ a[i+2] ^ u[i+3], ..., a[i] ^ a[i+1] ^ a[i+2] ^ ... ^ a[n-1] ^ u[n] } これを使えば,u[n], u[n-1], u[n-2], ... を漸次求めることができる. 0 箱が開封済みのときの Grundy 数である u[0] が答え. ★ このとき,開封済みの箱の Grundy 数が計算に不要なところがポイント! これのおかげで簡単に求められる. */ vll acc_a(n); // 累積 xor acc_a[0] = a[0]; REP(i, 1, n) acc_a[i] = acc_a[i - 1] ^ a[i]; auto range_xor = [&](ll l, ll r) -> ll { // 閉区間 if (l == 0) return acc_a[r]; return acc_a[r] ^ acc_a[l - 1]; }; vll u(n + 1); u[n] = 0; REPR(i, n - 1, 0) { // u[i] を求める set st; REP(r, i, n) st.insert(range_xor(i, r) ^ u[r + 1]); ll mex = 0; while (st.count(mex)) mex++; u[i] = mex; } pprint(u[0] > 0 ? "Takahashi" : "Takanashi"); } // entry point int main() { solve(); return 0; }