#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #define TE template #define TES template #define Z auto #define ep emplace_back #define eb emplace #define fi first #define se second #define bg begin #define ed end #define all(x) bg(x), ed(x) #define OV(a, b, c, d, e, ...) e #define FO1(a) for (int _ = 0; _ < (a); ++_) #define FO2(i, a) for (int i = 0; i < (a); ++i) #define FO3(i, a, b) for (int i = (a); i < (b); ++i) #define FO4(i, a, b, c) for (int i = (a); i < (b); i += (c)) #define FOR(...) OV(__VA_ARGS__, FO4, FO3, FO2, FO1)(__VA_ARGS__) #define FF1(a) for (int _ = (a) - 1; _ >= 0; --_) #define FF2(i, a) for (int i = (a) - 1; i >= 0; --i) #define FF3(i, a, b) for (int i = (b) - 1; i >= (a); --i) #define FF4(i, a, b, c) for (int i = (b) - 1; i >= (a); i -= (c)) #define FOR_R(...) OV(__VA_ARGS__, FF4, FF3, FF2, FF1)(__VA_ARGS__) #define FOR_subset(t, s) for (int t = (s); t > -1; t = (t == 0 ? -1 : (t - 1) & s)) #define sort ranges::sort using namespace std; TE using vc = vector; TE using vvc = vc>; TE using T1 = tuple; TE using T2 = tuple; TE using T3 = tuple; TE using T4 = tuple; TE using max_heap = priority_queue; TE using min_heap = priority_queue, greater>; using u8 = unsigned char; using uint = unsigned int; using ll = long long; using ull = unsigned long long; using ld = long double; using i128 = __int128; using u128 = __uint128_t; using f128 = __float128; using u16 = uint16_t; using PII = pair; using PLL = pair; #ifdef YRSD constexpr bool dbg = 1; #else constexpr bool dbg = 0; #endif TES concept nof = (not same_as and ...); istream &operator>>(istream &I, i128 &x) { static string s; I >> s; int f = s[0] == '-'; x = 0; const int N = (int)s.size(); FOR(i, f, N) x = x * 10 + s[i] - '0'; if (f) x = -x; return I; } ostream &operator<<(ostream &O, i128 x) { static string s; s.clear(); bool f = x < 0; if (f) x = -x; while (x) s += '0' + x % 10, x /= 10; if (s.empty()) s += '0'; if (f) s += '-'; reverse(all(s)); return O << s; } istream &operator>>(istream &I, f128 &x) { static string s; I >> s, x = stold(s); return I; } ostream &operator<<(ostream &O, const f128 x) { return O << ld(x); } template istream &operator>>(istream &I, tuple &t) { return apply([&I](Z &...s) { ((I >> s), ...); }, t), I; } template istream &operator>>(istream &I, pair &x) { return I >> x.fi >> x.se; } template ostream &operator<<(ostream &O, const pair &x) { return O << x.fi << ' ' << x.se; } TE requires requires(T &c) { begin(c); end(c); } and (not is_same_v, string>) istream &operator>>(istream &I, T &c) { for (Z &e : c) I >> e; return I; } TE ostream &operator<<(ostream &cout, const vc &a) { if (a.empty()) return cout; Z i = bg(a); cout << *i++; for (; i != ed(a); ++i) cout << ' ' << *i; return cout; } template requires(N > 0) ostream &operator<<(ostream &cout, const array &a) { cout << a[0]; FOR(i, 1, N) cout << ' ' << a[i]; return cout; } void IN() {} TE void IN(T &x, Z &...s) { cin >> x, IN(s...); } void print() { cout << '\n'; } TES void print(T &&x, S &&...y) { cout << x; if constexpr (sizeof...(S)) cout << ' '; print(forward(y)...); } void put() {} TES void put(T &&x, S &&...y) { cout << x; put(forward(y)...); } #define INT(...) int __VA_ARGS__; IN(__VA_ARGS__) #define UINT(...) uint __VA_ARGS__; IN(__VA_ARGS__) #define LL(...) ll __VA_ARGS__; IN(__VA_ARGS__) #define ULL(...) ull __VA_ARGS__; IN(__VA_ARGS__) #define I128(...) i128 __VA_ARGS__; IN(__VA_ARGS__) #define STR(...) string __VA_ARGS__; IN(__VA_ARGS__) #define CH(...) char __VA_ARGS__; IN(__VA_ARGS__) #define REAL(...) re __VA_ARGS__; IN(__VA_ARGS__) #define VEC(T, a, n) vc a(n); IN(a) void YES(bool o = 1) { print(o ? "YES" : "NO"); } void Yes(bool o = 1) { print(o ? "Yes" : "No"); } void yes(bool o = 1) { print(o ? "yes" : "no"); } void NO(bool o = 1) { YES(not o); } void No(bool o = 1) { Yes(not o); } void no(bool o = 1) { yes(not o); } void ALICE(bool o = 1) { print(o ? "ALICE" : "BOB"); } void Alice(bool o = 1) { print(o ? "Alice" : "Bob"); } void alice(bool o = 1) { print(o ? "alice" : "bob"); } void BOB(bool o = 1) { ALICE(not o); } void Bob(bool o = 1) { Alice(not o); } void bob(bool o = 1) { alice(not o); } void POSSIBLE(bool o = 1) { print(o ? "POSSIBLE" : "IMPOSSIBLE"); } void Possible(bool o = 1) { print(o ? "Possible" : "Impossible"); } void possible(bool o = 1) { print(o ? "possible" : "impossible"); } void IMPOSSIBLE(bool o = 1) { POSSIBLE(not o); } void Impossible(bool o = 1) { Possible(not o); } void impossible(bool o = 1) { possible(not o); } void TAK(bool o = 1) { print(o ? "TAK" : "NIE"); } void NIE(bool o = 1) { TAK(not o); } #if (__cplusplus >= 202002L) #include constexpr ld pi = numbers::pi_v; #endif TE constexpr T inf = numeric_limits::max(); template <> constexpr i128 inf = i128(inf) * 2'000'000'000'000'000'000; template constexpr pair inf> = {inf, inf}; TE constexpr static inline int pc(T x) { return popcount(make_unsigned_t(x)); } constexpr static inline ll si(const Z &a) { return a.size(); } void reverse(Z &a) { reverse(all(a)); } void unique(Z &a) { sort(a); a.erase(unique(all(a)), a.end()); } TE vc inverse(const vc &a) { int N = si(a); vc b(N, -1); FOR(i, N) if (a[i] != -1) b[a[i]] = i; return b; } Z QMAX(const Z &a) { return *max_element(all(a)); } Z QMIN(const Z &a) { return *min_element(all(a)); } TE Z QMAX(T l, T r) { return *max_element(l, r); } TE Z QMIN(T l, T r) { return *min_element(l, r); } constexpr bool chmax(Z &a, const Z &b) { return (a < b ? a = b, 1 : 0); } constexpr bool chmin(Z &a, const Z &b) { return (a > b ? a = b, 1 : 0); } vc argsort(const Z &a) { vc I(si(a)); iota(all(I), 0); sort(I, [&](int i, int k) { return a[i] < a[k] or (a[i] == a[k] and i < k); }); return I; } TE vc rearrange(const vc &a, const vc &I) { int N = si(I); vc b(N); FOR(i, N) b[i] = a[I[i]]; return b; } template vc pre_sum(const vc &a) { int N = si(a); vc c(N + 1); FOR(i, N) c[i + 1] = c[i] + a[i]; if (of == 0) c.erase(c.begin()); return c; } TE constexpr static int topbit(T x) { if (x == 0) return - 1; if constexpr (sizeof(T) <= 4) return 31 - __builtin_clz(x); else return 63 - __builtin_clzll(x); } TE constexpr static int lowbit(T x) { if (x == 0) return -1; if constexpr (sizeof(T) <= 4) return __builtin_ctz(x); else return __builtin_ctzll(x); } TE constexpr T floor(T x, T y) { return x / y - (x % y and (x ^ y) < 0); } TE constexpr T ceil(T x, T y) { return floor(x + y - 1, y); } TE constexpr T bmod(T x, T y) { return x - floor(x, y) * y; } TE constexpr pair divmod(T x, T y) { T q = floor(x, y); return pair{q, x - q * y}; } TE T SUM(const Z &v) { return accumulate(all(v), T()); } TE T SUM(Z l, Z r) { return accumulate(l, r, T()); } int lb(const Z &a, Z x) { return lower_bound(all(a), x) - a.begin(); } TE int lb(T l, T r, Z x) { return lower_bound(l, r, x) - l; } int ub(const Z &a, Z x) { return upper_bound(all(a), x) - a.begin(); } TE int ub(T l, T r, Z x) { return upper_bound(l, r, x) - l; } template ll bina(Z f, ll l, ll r) { if (ck) assert(f(l)); while (abs(l - r) > 1) { ll x = (r + l) >> 1; (f(x) ? l : r) = x; } return l; } TE T bina_real(Z f, T l, T r, int c = 100) { while (c--) { T x = (l + r) / 2; (f(x) ? l : r) = x; } return (l + r) / 2; } TE T pop(vc &a) { T x = a.back(); a.pop_back(); return x; } TE T pop(max_heap &q) { T x = q.top(); q.pop(); return x; } TE T pop(min_heap &q) { T x = q.top(); q.pop(); return x; } char pop(string &s) { char x = s.back(); s.pop_back(); return x; } void setp(int x) { cout << fixed << setprecision(x); } TE inline void sh(vc &a, int N, T b = {}) { a.resize(N, b); } TE requires (T::commute) struct bit_t : T { using X = T::X; using T::op, T::inv, T::unit; int N; vc a; bit_t() {} bit_t(int N) { build(N); } bit_t(int N, Z f) { build(N, f); } bit_t(const vc &a) { build(a); } void build(int M) { N = M, a.assign(N, unit()); } void build(const vc &a) { build(si(a), [&](int i) { return a[i]; }); } void build(int M, Z f) { N = M; a.resize(N); FOR(i, N) a[i] = f(i); FOR(i, 1, N + 1) { int k = i + (i & -i); if (k <= N) a[k - 1] = op(a[i - 1], a[k - 1]); } } void multiply(int i, X x) { for (; i < N; i |= i + 1) a[i] = op(a[i], x); } inline X prod(int i) { assert(i >= 0 and i <= N); X s = unit(); for (; i > 0; i -= i & -i) s = op(s, a[i - 1]); return s; } X prod(int l, int r) { assert(l <= r); return op(prod(r), inv(prod(l))); } X prod_all() { return prod(N); } void set(int i, X x) { multiply(i, op(inv(get(i)), x)); } X get(int i) { return prod(i, i + 1); } vc get_all() { vc s(N); FOR(i, N) s[i] = get(i); return s; } int max_right(Z f, int L = 0) { assert(f(unit())); X s = unit(); int i = L, k = [&]() -> int { while (1) { if (i & 1) s = op(s, inv(a[i - 1])), --i; if (i == 0) return topbit(N) + 1; int k = lowbit(i) - 1; if (i + (1 << k) > N) return k; X t = op(s, a[i + (1 << k) - 1]); if (not f(t)) return k; s = op(s, inv(a[i - 1])), i -= i & -i; } }(); while (k) { --k; if (i + (1 << k) - 1 < N) { X t = op(s, a[i + (1 << k) - 1]); if (f(t)) i += (1 << k), s = t; } } return i; } int kth(X k, int L = 0) { return max_right([&](X x) { return x <= k; }, L); } int min_left(Z f, int R) { assert(f(unit())); X s = unit(); int i = R, k = 0; while (i > 0 and f(s)) { s = op(s, a[i - 1]); k = lowbit(i); i -= i & -i; } if (f(s)) return assert(i == 0), 0; while (k) { --k; X t = op(s, inv(a[i + (1 << k) - 1])); if (not f(t)) i += (1 << k), s = t; } return i + 1; } }; TE struct monoid_add { using X = T; static constexpr inline X op(const X &x, const X &y) { return x + y; } static constexpr inline X inv(const X &x) { return -x; } static constexpr inline X pow(const X &x, ll n) { return X(n) * x; } static constexpr inline X unit() { return X(0); } static constexpr bool commute = 1; }; TE struct retsu { int N, M; vc a; retsu(int N, int M, T bs = T()) : N(N), M(M), a(N * M, bs) {} T* operator[](int i) { return a.data() + i * M; } const T* operator[](int i) const { return a.data() + i * M; } void fill(T x) { std::fill(all(a), x); } void build(int n, int m, T bs = T()) { N = n, M = m; a.assign(N * M, bs); } inline int id(int x, int y) const { return x * M + y; } retsu rol() const { retsu c(M, N); FOR(i, N) FOR(k, M) c[k][i] = a[id(i, k)]; return c; } retsu rol90() const { retsu c(M, N); FOR(i, N) FOR(k, M) c[k][i] = a[id(i, M - k - 1)]; return c; } void pres() { FOR(i, 1, N) FOR(k, M) a[id(i, k)] += a[id(i - 1, k)]; FOR(i, N) FOR(k, 1, M) a[id(i, k)] += a[id(i, k - 1)]; } inline T prod(int x, int y) const { return a[id(x, y)]; } inline T prod(int l, int r, int a, int b) const { return prod(r, b) + prod(l, a) - prod(l, b) - prod(r, a); } T max() const { return QMAX(a); } T min() const { return QMIN(a); } void reverse() { FOR(i, N) FOR(k, M >> 1) swap(a[i * M + k], a[i * M + M - k - 1]); } vc> to_v() const { vector c(N, vc(M)); FOR(i, N) FOR(k, M) c[i][k] = a[id(i, k)]; return c; } }; TE istream &operator>>(istream &cin, retsu &a) { for (Z &e : a.a) cin >> e; return cin; } TE ostream &operator<<(ostream &cout, const retsu &a) { FOR(i, a.N) { FOR(k, a.M) { if (k) cout << ' '; cout << a[i][k]; } if (i + 1 != a.N) cout << '\n'; } return cout; } #ifdef FIO TE inline void rd(retsu &a) { for (T &x : a.a) rd(x); } TE inline void wt(retsu &a) { FOR(i, a.N) { FOR(k, a.M) { if (k) wt(' '); wt(a[i][k]); } if (i != a.M) wt('\n'); } } #endif using DS = bit_t>; using A = retsu; A make(int N, int t) { A a(N, N); int c = ceil(N, 2); FOR(i, N) FOR(k, N) { int x = (i + k + c) % N, y = (2 * i + k + t) % N; a[i][k] = N * x + y + 1; } return a; } A make0(int N) { A a(N, N); FOR(i, N) FOR(k, N) { int g = i * N + k + 1; if (i % 4 == k % 4 or i % 4 + k % 4 == 3) g = N * N + 1 - g; a[i][k] = g; } return a; } A make1(int N) { int M = N >> 1, K = (N - 2) >> 2, mm = M * M; A a(N, N), b = make(M, 1); FOR(i, M) FOR(k, M) { int x = b[i][k]; a[i][k] = x; a[i][k + M] = x + 2 * mm; a[i + M][k] = x + 3 * mm; a[i + M][k + M] = x + mm; } FOR(i, M) { FOR(k, K) swap(a[i][k], a[i + M][k]); FOR(k, N - K + 1, N) swap(a[i][k], a[i + M][k]); } swap(a[K][0], a[K + M][0]); swap(a[K][K], a[K + M][K]); return a; } void Yorisou() { INT(N); A a(N, N); IN(a); if (N == 1) return possible(), print(1); if (N == 2) return impossible(); Z f = [&](const A &a, int w) -> bool { vc v; int t = -1; FOR(i, N) FOR(k, N) { if (a[i][k] == w) t = i; else v.ep(a[i][k]); } DS bit(N * N + 2); int in = 0, se = 0; for (int x : v) { x %= N * N; int g = se - bit.prod(x); in ^= g & 1; bit.multiply(x, 1); ++se; } if (N & 1) return in; return in ^ ((N - t) & 1); }; int st = f(a, 0); vc c; if (N & 1) { if (N == 3) { A a = make(3, 1); c.ep(a); a.reverse(); c.ep(a); } else { c.ep(make(N, 1)); c.ep(make(N, 1 + gcd(N, 3))); } } else { A a = (N & 3 ? make1(N) : make0(N)); c.ep(a); a.reverse(); c.ep(a); } for (Z &a : c) { int in = f(a, N * N); if (in == st) { possible(); print(a); return; } } impossible(); } constexpr int tests = 0, fl = 0, DB = 10; int main() { cin.tie(0)->sync_with_stdio(0); int T = 1; if (fl) cerr.tie(0); if (tests and not fl) IN(T); for (int i = 0; i < T or fl; ++i) { Yorisou(); if (fl and i % DB == 0) cerr << "Case: " << i << '\n'; } return 0; }