#pragma GCC optimize("O3") #pragma GCC optimize("unroll-loops") #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; #define all(a) (a).begin(), (a).end() #define rep(i, n) for (ll i = 0; i < (n); i++) #define For(i, a, b) for (ll i = (a); i < (b); i++) #define debug(...) cerr << __LINE__ << " | ", debug_out(#__VA_ARGS__, __VA_ARGS__) #define test_only(x, y, ...) do { if ((x) != (y)) { debug(x, y, ##__VA_ARGS__); } } while (0) #define test_same(x, y, ...) do { if ((x) != (y)) { debug(x, y, ##__VA_ARGS__); abort(); } } while (0) typedef long long ll; typedef unsigned int uint; typedef unsigned long long ull; typedef long double ld; template using P = pair; template using pri_l = priority_queue; template using pri_s = priority_queue, greater>; constexpr int inf = 1000000010; constexpr ll INF = 1000000000000000010; constexpr int mod1e9 = 1000000007; constexpr int mod998 = 998244353; constexpr ld eps = 1e-12; constexpr ld pi = 3.141592653589793238; constexpr ll ten(int n) { return n ? 10 * ten(n - 1) : 1; }; int dx[] = { 1,0,-1,0,1,1,-1,-1,0 }; int dy[] = { 0,1,0,-1,1,-1,1,-1,0 }; ll safe_mul(ll a, ll b, ll LIM = INF) { return (b != 0 && a > LIM / b ? LIM : a * b); } void fail() { cout << "-1\n"; exit(0); } void no() { cout << "No\n"; exit(0); } template void er(T a) { cout << a << '\n'; exit(0); } template inline bool chmax(T& a, const U& b) { if (a < b) { a = b; return true; } return false; } template inline bool chmin(T& a, const U& b) { if (a > b) { a = b; return true; } return false; } template ostream& operator << (ostream& s, const pair& p) { s << p.first << ' ' << p.second; return s; } template istream& operator >>(istream& s, vector& v) { for (auto& e : v) s >> e; return s; } template ostream& operator <<(ostream& s, const vector& v) { for (auto& e : v) s << e << ' '; return s; } struct fastio { fastio() { cin.tie(0); cout.tie(0); ios::sync_with_stdio(false); cout << fixed << setprecision(20); cerr << fixed << setprecision(20); } }fastio_; namespace rdv { random_device seed_gen; mt19937_64 engine(seed_gen()); ll rnum(ll r) { return engine() % r; } // [0, r) ll rnum(ll l, ll r) { return rnum(r - l) + l; } // [l, r) ll rng(ll l, ll r) { return rnum(l, r + 1); } // [l, r] double rng01() { return engine() * pow(2, -64); } template void shuf(vector& v) { shuffle(all(v), engine); } void shuf(string& s) { shuffle(all(s), engine); } } using namespace rdv; template vector compress(vector& v) { int n = ssize(v); vector tmp = v; sort(tmp.begin(), tmp.end()); tmp.erase(unique(tmp.begin(), tmp.end()), tmp.end()); vector res(n); for (int i = 0; i < n; i++) res[i] = lower_bound(tmp.begin(), tmp.end(), v[i]) - tmp.begin(); return res; } #ifdef _MSC_VER using lint = ll; #else using lint = __int128_t; #endif #include using namespace atcoder; constexpr ll mod = mod998; using mint = static_modint; istream& operator >>(istream& s, mint& m) { ll y; s >> y; m = y; return s; } istream& operator >>(istream& s, vector& v) { for (auto& e : v) { ll y; s >> y; e = y; } return s; } ostream& operator <<(ostream& s, const mint& m) { return s << m.val(); } ostream& operator <<(ostream& s, const vector& v) { for (auto& e : v) s << e.val() << ' '; return s; } void debug_out(const char*) { cerr << "\n"; } template void debug_out(const char* names, T value, Args... args) { while (*names == ' ') ++names; const char* comma = strchr(names, ','); if (!comma) { cerr << names << ":" << value << "\n"; } else { cerr.write(names, comma - names) << ":" << value << ", "; debug_out(comma + 1, args...); } } vector fac, inv, facinv; void modcalc(int n) { assert(fac.empty()); assert(inv.empty()); assert(facinv.empty()); fac.resize(n); inv.resize(n); facinv.resize(n); fac[0] = 1; fac[1] = 1; inv[1] = 1; facinv[0] = 1; facinv[1] = 1; for (ll i = 2; i < n; i++) { fac[i] = fac[i - 1] * i; inv[i] = -inv[mod % i] * (mod / i); facinv[i] = facinv[i - 1] * inv[i]; } } mint comb(ll n, ll k) { if (n < 0 or k < 0 or n < k) return 0; return fac[n] * facinv[k] * facinv[n - k]; } mint perm(ll n, ll k) { if (n < 0 or k < 0 or n < k) return 0; return fac[n] * facinv[n - k]; } mint hom(ll n, ll k) { if (n < 0 or k < 0 or (n == 0 && k > 0)) return 0; if (n == 0 && k == 0) return 1; return fac[n + k - 1] * facinv[k] * facinv[n - 1]; } int main() { [[maybe_unused]] bool DEBUG = false; int TEST = 1; cin >> TEST; while (TEST--) { int n; if (DEBUG) n = rng(1, 3); else cin >> n; vector x(n), y(n), z(2 * n - 1); if (DEBUG) { rep(i, n) { x[i] = rng(0, 7); y[i] = rng(0, 7); } rep(i, 2 * n - 1) z[i] = rng(0, 15); } else cin >> x >> y >> z; vector a(n, vector(n)); if (n == 1) { if (x[0] == y[0] && y[0] == z[0]) { cout << x[0] << '\n'; continue; } else { cout << "-1\n"; continue; } } assert(n >= 2); bool bad = false; rep(d, 30) { vector f(n, vector(n, -1)); bool and_1 = false, or_0 = false; rep(i, n) if (x[i] >> d & 1) and_1 = true; rep(i, n) if (!(y[i] >> d & 1)) or_0 = true; if (and_1 && or_0) { bad = true; break; } if (and_1) { rep(i, n) { if (x[i] >> d & 1) { rep(j, n) f[i][j] = 1; } } int p = -1, q = -1; rep(i, n) { if (!(x[i] >> d & 1)) { if (p == -1) p = i; else if (q == -1) q = i; else rep(j, n) f[i][j] = 0; } } // p, q 行目以外は適当に埋める // k = n-1 はどちらでも発生 // p, q 行目を両方 0 にできるなら終わり // 目標が 1 のとき、片方しか 0 にできない // 両方全 1 のときにのみ困っている // 1, 1 という埋め方をしないというルールを付けると、p=0, q=n-1 以外では困らない // 逆に p=0,q=n-1 のときは選択の余地がないからいいはず vector ord; rep(k, 2 * n - 1) { if (k != n - 1) ord.push_back(k); } ord.push_back(n - 1); for (int k : ord) { vector> pos; int sum = z[k] >> d & 1; rep(i, n) { int j = k - i; if (!(0 <= j && j < n)) continue; if (f[i][j] != -1) sum ^= f[i][j]; else pos.push_back({ i, j }); } if (pos.empty() && sum != 0) bad = true; assert(ssize(pos) <= 2); if (ssize(pos) == 1) { f[pos[0].first][pos[0].second] = sum; } else if (ssize(pos) == 2) { if (sum == 0) { f[pos[0].first][pos[0].second] = 0; f[pos[1].first][pos[1].second] = 0; } else { bool have_0_p = false, have_0_q = false; rep(j, n) { if (f[p][j] == 0) have_0_p = true; if (f[q][j] == 0) have_0_q = true; } if (!have_0_p && !have_0_q) { if (k == n - 1) { assert(p == 0 && q == n - 1); bad = true; } else { f[pos[0].first][pos[0].second] = 0; f[pos[1].first][pos[1].second] = 1; } } else if (have_0_p) { f[pos[0].first][pos[0].second] = 1; f[pos[1].first][pos[1].second] = 0; } else { f[pos[0].first][pos[0].second] = 0; f[pos[1].first][pos[1].second] = 1; } } } } } else if (or_0) { rep(j, n) { if (!(y[j] >> d & 1)) { rep(i, n) f[i][j] = 0; } } int p = -1, q = -1; rep(j, n) { if (y[j] >> d & 1) { if (p == -1) p = j; else if (q == -1) q = j; else rep(i, n) f[i][j] = 1; } } vector ord; rep(k, 2 * n - 1) { if (k != n - 1) ord.push_back(k); } ord.push_back(n - 1); for (int k : ord) { vector> pos; int sum = z[k] >> d & 1; rep(j, n) { int i = k - j; if (!(0 <= i && i < n)) continue; if (f[i][j] != -1) sum ^= f[i][j]; else pos.push_back({ i, j }); } if (pos.empty() && sum != 0) bad = true; assert(ssize(pos) <= 2); if (ssize(pos) == 1) { f[pos[0].first][pos[0].second] = sum; } else if (ssize(pos) == 2) { if (sum == 0) { f[pos[0].first][pos[0].second] = 1; f[pos[1].first][pos[1].second] = 1; } else { bool have_1_p = false, have_1_q = false; rep(i, n) { if (f[i][p] == 1) have_1_p = true; if (f[i][q] == 1) have_1_q = true; } if (!have_1_p && !have_1_q) { if (k == n - 1) { assert(p == 0 && q == n - 1); bad = true; } else { f[pos[0].first][pos[0].second] = 1; f[pos[1].first][pos[1].second] = 0; } } else if (have_1_p) { f[pos[0].first][pos[0].second] = 0; f[pos[1].first][pos[1].second] = 1; } else { f[pos[0].first][pos[0].second] = 1; f[pos[1].first][pos[1].second] = 0; } } } } } else { if (n == 2) { int z0 = z[0] >> d & 1; int z1 = z[1] >> d & 1; int z2 = z[2] >> d & 1; if (z0 == 0 && z1 == 0 && z2 == 0) { f[0][0] = 0, f[0][1] = 1, f[1][0] = 1, f[1][1] = 0; } else if (z0 == 1 && z1 == 0 && z2 == 1) { f[0][0] = 1, f[0][1] = 0, f[1][0] = 0, f[1][1] = 1; } else { bad = true; break; } } else if (n == 3) { f[0][0] = z[0] >> d & 1; f[2][2] = z[4] >> d & 1; bool valid = false; rep(bt, 16) { int f01 = bt >> 0 & 1; int f02 = bt >> 1 & 1; int f11 = bt >> 2 & 1; int f12 = bt >> 3 & 1; int f00 = f[0][0], f22 = f[2][2]; int f10 = (z[1] >> d & 1) ^ f01; int f20 = (z[2] >> d & 1) ^ f02 ^ f11; int f21 = (z[3] >> d & 1) ^ f12; bool x0 = f00 == 0 or f01 == 0 or f02 == 0; bool x1 = f10 == 0 or f11 == 0 or f12 == 0; bool x2 = f20 == 0 or f21 == 0 or f22 == 0; bool y0 = f00 == 1 or f10 == 1 or f20 == 1; bool y1 = f01 == 1 or f11 == 1 or f21 == 1; bool y2 = f02 == 1 or f12 == 1 or f22 == 1; if (x0 && x1 && x2 && y0 && y1 && y2) { valid = true; f[0][1] = f01, f[0][2] = f02; f[1][0] = f10, f[1][1] = f11, f[1][2] = f12; f[2][0] = f20, f[2][1] = f21; break; } } if (!valid) bad = true; } else if (n >= 4) { rep(i, n) { f[(i + 1) % n][i] = 0; f[(i + 2) % n][i] = 1; } rep(k, 2 * n - 1) { int sum = z[k] >> d & 1; vector> pos; rep(i, n) { int j = k - i; if (!(0 <= j && j < n)) continue; if (f[i][j] != -1) sum ^= f[i][j]; else pos.push_back({ i, j }); } assert(!pos.empty()); f[pos[0].first][pos[0].second] = sum; For(i, 1, ssize(pos)) f[pos[i].first][pos[i].second] = 0; } } } if (bad) break; // debug(f); rep(i, n) rep(j, n) assert(f[i][j] != -1); rep(i, n) rep(j, n) a[i][j] += f[i][j] << d; } { vector X(n, (1 << 30) - 1), Y(n), Z(2 * n - 1); rep(i, n) rep(j, n) { X[i] &= a[i][j]; Y[j] |= a[i][j]; Z[i + j] ^= a[i][j]; } if (!(x == X && y == Y && z == Z)) bad = true; } if (bad) cout << "-1\n"; else { rep(i, n) rep(j, n) cout << a[i][j] << " \n"[j == n - 1]; if (DEBUG) { vector X(n, (1 << 30) - 1), Y(n), Z(2 * n - 1); rep(i, n) rep(j, n) { X[i] &= a[i][j]; Y[j] |= a[i][j]; Z[i + j] ^= a[i][j]; } debug(x, y, z, X, Y, Z); assert(x == X && y == Y && z == Z); } } } }