2014
11-30

# 3D Convex Hull

There are N points in 3D-space which make up a 3D-Convex hull*. How many faces does the 3D-convexhull have? It is guaranteed that all the points are not in the same plane.

In case you don’t know the definition of convex hull, here we give you a clarification from Wikipedia:
*Convex hull: In mathematics, the convex hull, for a set of points X in a real vector space V, is the minimal convex set containing X.

There are several test cases. In each case the first line contains an integer N indicates the number of 3D-points (3< N <= 300), and then N lines follow, each line contains three numbers x, y, z (between -10000 and 10000) indicate the 3d-position of a point.

There are several test cases. In each case the first line contains an integer N indicates the number of 3D-points (3< N <= 300), and then N lines follow, each line contains three numbers x, y, z (between -10000 and 10000) indicate the 3d-position of a point.

7
1 1 0
1 -1 0
-1 1 0
-1 -1 0
0 0 1
0 0 0
0 0 -0.1
7
1 1 0
1 -1 0
-1 1 0
-1 -1 0
0 0 1
0 0 0
0 0 0.1

8
5

/** Micro Mezzo Macro Flation -- Overheated Economy ., Last Update: Dec. 4th 2012 **/ //{

#define LOCAL

#include <functional>
#include <algorithm>
#include <iostream>
#include <fstream>
#include <sstream>
#include <iomanip>
#include <numeric>
#include <cstring>
#include <cassert>
#include <cstdio>
#include <string>
#include <vector>
#include <bitset>
#include <queue>
#include <stack>
#include <cmath>
#include <ctime>
#include <list>
#include <set>
#include <map>

using namespace std;

#define REP(i, n) for (int i=0;i<int(n);++i)
#define FOR(i, a, b) for (int i=int(a);i<int(b);++i)
#define DWN(i, b, a) for (int i=int(b-1);i>=int(a);--i)
#define REP_1(i, n) for (int i=1;i<=int(n);++i)
#define FOR_1(i, a, b) for (int i=int(a);i<=int(b);++i)
#define DWN_1(i, b, a) for (int i=int(b);i>=int(a);--i)
#define REP_C(i, n) for (int n____=int(n),i=0;i<n____;++i)
#define FOR_C(i, a, b) for (int b____=int(b),i=a;i<b____;++i)
#define DWN_C(i, b, a) for (int a____=int(a),i=b-1;i>=a____;--i)
#define REP_N(i, n) for (i=0;i<int(n);++i)
#define FOR_N(i, a, b) for (i=int(a);i<int(b);++i)
#define DWN_N(i, b, a) for (i=int(b-1);i>=int(a);--i)
#define REP_1_C(i, n) for (int n____=int(n),i=1;i<=n____;++i)
#define FOR_1_C(i, a, b) for (int b____=int(b),i=a;i<=b____;++i)
#define DWN_1_C(i, b, a) for (int a____=int(a),i=b;i>=a____;--i)
#define REP_1_N(i, n) for (i=1;i<=int(n);++i)
#define FOR_1_N(i, a, b) for (i=int(a);i<=int(b);++i)
#define DWN_1_N(i, b, a) for (i=int(b);i>=int(a);--i)
#define REP_C_N(i, n) for (int n____=(i=0,int(n));i<n____;++i)
#define FOR_C_N(i, a, b) for (int b____=(i=0,int(b);i<b____;++i)
#define DWN_C_N(i, b, a) for (int a____=(i=b-1,int(a));i>=a____;--i)
#define REP_1_C_N(i, n) for (int n____=(i=1,int(n));i<=n____;++i)
#define FOR_1_C_N(i, a, b) for (int b____=(i=1,int(b);i<=b____;++i)
#define DWN_1_C_N(i, b, a) for (int a____=(i=b,int(a));i>=a____;--i)

//#define ECH(it, A) for (typeof(A.begin()) it=A.begin(); it != A.end(); ++it)
#define ECH(it, A) for (__typeof(A.begin()) it=A.begin(); it != A.end(); ++it)
#define REP_S(it, str) for (char*it=str;*it;++it)
#define REP_G(it, u) for (int it=hd[u];it;it=suc[it])
#define DO(n) for ( int ____n ## __line__ = n; ____n ## __line__ -- ; )
#define REP_2(i, j, n, m) REP(i, n) REP(j, m)
#define REP_2_1(i, j, n, m) REP_1(i, n) REP_1(j, m)
#define REP_3(i, j, k, n, m, l) REP(i, n) REP(j, m) REP(k, l)
#define REP_3_1(i, j, k, n, m, l) REP_1(i, n) REP_1(j, m) REP_1(k, l)

#define ALL(A) A.begin(), A.end()
#define LLA(A) A.rbegin(), A.rend()
#define CPY(A, B) memcpy(A, B, sizeof(A))
#define INS(A, P, B) A.insert(A.begin() + P, B)
#define ERS(A, P) A.erase(A.begin() + P)
#define BSC(A, X) find(ALL(A), X) // != A.end()
#define CTN(T, x) (T.find(x) != T.end())
#define SZ(A) int(A.size())
#define PB push_back
#define MP(A, B) make_pair(A, B)
#define PTT pair<T, T>
#define fi first
#define se second

#define Rush for(int ____T=RD(); ____T--;)

//#pragma GCC optimize ("O2")
#define Ruby system("ruby main.rb")
#define Python system("python main.py")
#define Pascal system("fpc main.pas")

typedef long long LL;
//typedef long double DB;
typedef double DB;
typedef unsigned UINT;
typedef unsigned long long ULL;

typedef vector<int> VI;
typedef vector<char> VC;
typedef vector<string> VS;
typedef vector<LL> VL;
typedef vector<DB> VD;
typedef set<int> SI;
typedef set<string> SS;
typedef map<int, int> MII;
typedef map<string, int> MSI;
typedef pair<int, int> PII;
typedef pair<LL, LL> PLL;
typedef vector<PII> VII;
typedef vector<VI> VVI;
typedef vector<VII> VVII;

template<class T> inline T& RD(T &);
template<class T> inline void OT(const T &);
inline LL RD(){LL x; return RD(x);}
inline DB& RF(DB &x){scanf("%lf", &x); return x;}
inline DB RF(){DB x; return RF(x);}
inline char* RS(char *s);
inline char& RC(char &c);
inline char RC();
inline char& RC(char &c){scanf(" %c", &c); return c;}
inline char RC(){char c; return RC(c);}
//inline char& RC(char &c){c = getchar(); return c;}
//inline char RC(){return getchar();}

template<class T0, class T1> inline T0& RD(T0 &x0, T1 &x1){RD(x0), RD(x1); return x0;}
template<class T0, class T1, class T2> inline T0& RD(T0 &x0, T1 &x1, T2 &x2){RD(x0), RD(x1), RD(x2); return x0;}
template<class T0, class T1, class T2, class T3> inline T0& RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3){RD(x0), RD(x1), RD(x2), RD(x3); return x0;}
template<class T0, class T1, class T2, class T3, class T4> inline T0& RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4); return x0;}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline T0& RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5); return x0;}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline T0& RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5, T6 &x6){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5), RD(x6); return x0;}
template<class T0, class T1> inline void OT(const T0 &x0, const T1 &x1){OT(x0), OT(x1);}
template<class T0, class T1, class T2> inline void OT(const T0 &x0, const T1 &x1, const T2 &x2){OT(x0), OT(x1), OT(x2);}
template<class T0, class T1, class T2, class T3> inline void OT(const T0 &x0, const T1 &x1, const T2 &x2, const T3 &x3){OT(x0), OT(x1), OT(x2), OT(x3);}
template<class T0, class T1, class T2, class T3, class T4> inline void OT(const T0 &x0, const T1 &x1, const T2 &x2, const T3 &x3, const T4 &x4){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void OT(const T0 &x0, const T1 &x1, const T2 &x2, const T3 &x3, const T4 &x4, const T5 &x5){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void OT(const T0 &x0, const T1 &x1, const T2 &x2, const T3 &x3, const T4 &x4, const T5 &x5, const T6 &x6){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5), OT(x6);}
inline char& RC(char &a, char &b){RC(a), RC(b); return a;}
inline char& RC(char &a, char &b, char &c){RC(a), RC(b), RC(c); return a;}
inline char& RC(char &a, char &b, char &c, char &d){RC(a), RC(b), RC(c), RC(d); return a;}
inline DB& RF(DB &a, DB &b){RF(a), RF(b); return a;}
inline DB& RF(DB &a, DB &b, DB &c){RF(a), RF(b), RF(c); return a;}
inline DB& RF(DB &a, DB &b, DB &c, DB &d){RF(a), RF(b), RF(c), RF(d); return a;}
inline void RS(char *s1, char *s2){RS(s1), RS(s2);}
inline void RS(char *s1, char *s2, char *s3){RS(s1), RS(s2), RS(s3);}

template<class T> inline void RST(T &A){memset(A, 0, sizeof(A));}
template<class T0, class T1> inline void RST(T0 &A0, T1 &A1){RST(A0), RST(A1);}

template<class T> inline T& SRT(T &A){sort(ALL(A)); return A;}
template<class T, class C> inline T& SRT(T &A, C B){sort(ALL(A), B); return A;}

//}

/** Constant List .. **/ //{

const int MOD = 1000000007;
//int MOD = 99990001;
const int INF = 0x3f3f3f3f;
const LL INFF = 1LL << 60;
const DB EPS = 1e-9;
const DB OO = 1e15;
const DB PI = acos(-1.0); //M_PI;

const int dx[] = {-1, 0, 1, 0};
const int dy[] = {0, 1, 0, -1};

//}

/** Add On .. **/ //{
// <<= '0. Nichi Joo ., //{
template<class T> inline void checkMin(T &a,const T b){if (b<a) a=b;}
template<class T> inline void checkMax(T &a,const T b){if (a<b) a=b;}
template<class T> inline void checkMin(T &a, T &b, const T x){checkMin(a, x), checkMin(b, x);}
template<class T> inline void checkMax(T &a, T &b, const T x){checkMax(a, x), checkMax(b, x);}
template <class T, class C> inline void checkMin(T& a, const T b, C c){if (c(b,a)) a = b;}
template <class T, class C> inline void checkMax(T& a, const T b, C c){if (c(a,b)) a = b;}
template<class T> inline T min(T a, T b, T c){return min(min(a, b), c);}
template<class T> inline T max(T a, T b, T c){return max(max(a, b), c);}
template<class T> inline T min(T a, T b, T c, T d){return min(min(a, b), min(c, d));}
template<class T> inline T max(T a, T b, T c, T d){return max(max(a, b), max(c, d));}
template<class T> inline T sqr(T a){return a*a;}
template<class T> inline T cub(T a){return a*a*a;}
inline int Ceil(int x, int y){return (x - 1) / y + 1;}
//}
namespace BO{

inline bool _1(int x, int i){return bool(x&1<<i);}
inline bool _1(LL x, int i){return bool(x&1LL<<i);}
inline LL _1(int i){return 1LL<<i;}
inline LL _U(int i){return _1(i) - 1;};

inline int reverse_bits(int x){
x = ((x >> 1) & 0x55555555) | ((x << 1) & 0xaaaaaaaa);
x = ((x >> 2) & 0x33333333) | ((x << 2) & 0xcccccccc);
x = ((x >> 4) & 0x0f0f0f0f) | ((x << 4) & 0xf0f0f0f0);
x = ((x >> 8) & 0x00ff00ff) | ((x << 8) & 0xff00ff00);
x = ((x >>16) & 0x0000ffff) | ((x <<16) & 0xffff0000);
return x;
}

inline LL reverse_bits(LL x){
x = ((x >> 1) & 0x5555555555555555LL) | ((x << 1) & 0xaaaaaaaaaaaaaaaaLL);
x = ((x >> 2) & 0x3333333333333333LL) | ((x << 2) & 0xccccccccccccccccLL);
x = ((x >> 4) & 0x0f0f0f0f0f0f0f0fLL) | ((x << 4) & 0xf0f0f0f0f0f0f0f0LL);
x = ((x >> 8) & 0x00ff00ff00ff00ffLL) | ((x << 8) & 0xff00ff00ff00ff00LL);
x = ((x >>16) & 0x0000ffff0000ffffLL) | ((x <<16) & 0xffff0000ffff0000LL);
x = ((x >>32) & 0x00000000ffffffffLL) | ((x <<32) & 0xffffffff00000000LL);
return x;
}

template<class T> inline bool odd(T x){return x&1;}
template<class T> inline bool even(T x){return !odd(x);}
template<class T> inline T low_bit(T x) {return x & -x;}
template<class T> inline T high_bit(T x) {T p = low_bit(x);while (p != x) x -= p, p = low_bit(x);return p;}
template<class T> inline T cover_bit(T x){T p = 1; while (p < x) p <<= 1;return p;}

inline int low_idx(int x){return __builtin_ffs(x);}
inline int low_idx(LL x){return __builtin_ffsll(x);}
inline int high_idx(int x){return low_idx(reverse_bits(x));}
inline int high_idx(LL x){return low_idx(reverse_bits(x));}
inline int clz(int x){return __builtin_clz(x);}
inline int clz(LL x){return __builtin_clzll(x);}
inline int ctz(int x){return __builtin_ctz(x);}
inline int ctz(LL x){return __builtin_ctzll(x);}
inline int parity(int x){return __builtin_parity(x);}
inline int parity(LL x){return __builtin_parityll(x);}
inline int lg2(int a){return 31 - clz(a);}
inline int lg2(LL a){return 63 - clz(a);}
inline int count_bits(int x){return __builtin_popcount(x);}
inline int count_bits(LL x){return __builtin_popcountll(x);}

} using namespace BO;//}
namespace NT{
inline LL __lcm(LL a, LL b){return a*b/__gcd(a,b);}
inline void INC(int &a, int b){a += b; if (a >= MOD) a -= MOD;}
inline int sum(int a, int b){a += b; if (a >= MOD) a -= MOD; return a;}
inline void DEC(int &a, int b){a -= b; if (a < 0) a += MOD;}
inline int dff(int a, int b){a -= b; if (a < 0) a += MOD; return a;}
inline void MUL(int &a, int b){a = (LL)a * b % MOD;}
inline int pdt(int a, int b){return (LL)a * b % MOD;}

inline int sum(int a, int b, int c){return sum(sum(a, b), c);}
inline int sum(int a, int b, int c, int d){return sum(sum(a, b), sum(c, d));}
inline int pdt(int a, int b, int c){return pdt(pdt(a, b), c);}
inline int pdt(int a, int b, int c, int d){return pdt(pdt(pdt(a, b), c), d);}

inline int pow(int a, int b){
int c(1); while (b){
if (b&1) MUL(c, a);
MUL(a, a), b >>= 1;
}
return c;
}

inline int pow(int a, LL b){
int c(1); while (b){
if (b&1) MUL(c, a);
MUL(a, a), b >>= 1;
}
return c;
}

template<class T> inline T pow(T a, LL b){
T c(1); while (b){
if (b&1) c *= a;
a *= a, b >>= 1;
}
return c;
}

inline int _I(int b){
int a = MOD, x1 = 0, x2 = 1, q;
while (true){
q = a / b, a %= b;
if (!a) return (x2 + MOD) % MOD;
DEC(x1, pdt(q, x2));

q = b / a, b %= a;
if (!b) return (x1 + MOD) % MOD;
DEC(x2, pdt(q, x1));
}
}

inline void DIV(int &a, int b){MUL(a, _I(b));}
inline int qtt(int a, int b){return pdt(a, _I(b));}

inline int phi(int n){
int res = n; for (int i=2;sqr(i)<=n;++i) if (!(n%i)){
DEC(res, qtt(res, i));
do{n /= i;} while(!(n%i));
}
if (n != 1)
DEC(res, qtt(res, n));
return res;
}

} using namespace NT;//}

namespace SL{
namespace KMP{

void calc_pi(const char *P, int n, int *pi){
for (int i = 1, j = pi[0] = -1; i < n; ++i){
while (j >= 0 && P[i] != P[j+1]) j = pi[j];
if (P[i] == P[j+1]) ++j;
pi[i] = j;
}
//REP(i, n) cout << pi[i] << " "; cout << endl;
}

bool run(const char *T, int n, const char *P, int m, const int *pi){
for (int i = 0, j = -1; i < n; ++i){
while (j >= 0 && T[i] != P[j+1]) j = pi[j];
if (T[i] == P[j+1]) ++j;
if (j == m - 1) return true;
}
return false;
}

} //using namespace KMP;

namespace Z{
void calc_z(const char *P, int n, int *z){

z[0] = n;

for (int i = 1, l = 0, r = 0; i < n; ++i){
if (i > r){
for(l = r = i; r < n && P[r] == P[r - l];) ++r;
z[i] = r - l, --r;
}
else {
if (z[i - l] < r - i + 1) z[i] = z[i - l];
else {
for (l = i;r < n && P[r] == P[r - l];) ++r;
z[i] = r - l, --r;
}
}
}

//REP(i, n) cout << z[i] << " "; cout << endl;
}

int run(const char *T, int n, const char *P, int m, const int *z){

int ex; REP_C_N(ex, min(n, m)) if (T[ex] != P[ex]) break;

int res = ex == m;

for (int i = 1, l = 0, r = 0; i < n; ++i){
if (i > r){
for (l = r = i; r < n && T[r] == P[r - l];) ++r;
ex = r - l, --r;
}
else {
if (z[i - l] < r - i + 1) ex = z[i - l];
else {
for (l = i; r < n && T[r] == P[r - l];) ++r;
ex = r - l, --r;
}
}
if (ex == m) ++res;
}

return res;
}
} //using namespace Z;

void Manacher(char s[], int n, int p[]){
const int NN = 0;
static char ss[NN*2+2]; int nn = 2*n+2;
ss[0] = '\$', ss[nn-1] = '#', ss[nn] = 0;
REP(i, n) ss[i*2+1] ='#', ss[i*2+2] = s[i];
int mx = 0, id = 0; FOR(i, 1, nn){
p[i] = mx > i ? min(p[2*id-i], mx - i) : 1;
while (ss[i+p[i]] == ss[i-p[i]]) ++p[i];
if (i + p[i] > mx) mx = i + p[i], id = i;
}
}

} //using namespace SL;//}
// <<= '9. Comutational Geometry .,//{
namespace CG{

struct Po; struct Line; struct Seg;

inline int sgn(DB x){return x < -EPS ? -1 : x > EPS;}
inline int sgn(DB x, DB y){return sgn(x - y);}

struct Po{
DB x, y; Po(DB _x=0, DB _y=0):x(_x), y(_y){}
friend istream& operator >>(istream& in, Po &p){return in >> p.x >> p.y;}
friend ostream& operator <<(ostream& out, Po p){return out << "(" << p.x << ", " << p.y << ")";}
bool operator ==(const Po& r)const{return !sgn(x-r.x) && !sgn(y-r.y);};
bool operator !=(const Po& r)const{return sgn(x-r.x) || sgn(y-r.y);}
Po operator +(const Po& r)const{return Po(x+r.x, y+r.y);}
Po operator -(const Po& r)const{return Po(x+r.x, y+r.y);}
Po operator *(DB k)const{return Po(x*k,y*k);}
Po operator /(DB k)const{return Po(x/k,y/k);}
DB operator *(const Po&) const;
DB operator ^(const Po&) const;

bool operator <(const Po &r) const{return sgn(x,r.x)<0||!sgn(x,r.x)&&sgn(y,r.y)<0;}
Po operator -()const{return Po(-x,-y);}
Po& operator +=(const Po &r){x+=r.x,y+=r.y;return *this;}
Po& operator -=(const Po &r){x-=r.x,y-=r.y;return *this;}
Po& operator *=(DB k){x*=k,y*=k;return*this;}
Po& operator /=(DB k){x/=k,y/=k;return*this;}

DB length_sqr()const{return sqr(x)+sqr(y);}
DB length()const{return sqrt(length_sqr());}
Po unit()const{return *this/length();}
bool dgt()const{return !sgn(x)&&!sgn(y);}
DB atan()const{return atan2(y,x);}
void input(){RF(x,y);}
};

Po operator *(DB k, Po a){return a * k;}

struct Line{
Po a, b;
Line(DB x0=0, DB y0=0, DB x1=0, DB y1=0):a(Po(x0, y0)), b(Po(x1, y1)){}
Line(const Po &a, const Po &b):a(a), b(b){}
Line(const Line &l):a(l.a), b(l.b){}

friend ostream& operator <<(ostream& out, Line p){return out << p.a << "-" << p.b;}
Line operator +(Po x)const{return Line(a + x, b + x);}
DB length()const{return (b-a).length();}
bool dgt()const{return (b-a).dgt();}
void input(){a.input(), b.input();}
void getequation(DB, DB, DB) const;
};

struct Seg: Line{
};

#define innerProduct dot
#define scalarProduct dot
#define dotProduct dot
#define outerProduct det
#define crossProduct det

inline DB dot(const DB &x1, const DB &y1, const DB &x2, const DB &y2){return x1 * x2 + y1 * y2;}
inline DB dot(const Po &a, const Po &b){return dot(a.x, a.y, b.x, b.y);}
inline DB dot(const Po &p0, const Po &p1, const Po &p2){return dot(p1 - p0, p2 - p0);}
inline DB dot(const Po &o, const Line &l){return dot(o, l.a, l.b);}
inline DB dot(const Line &l, const Po &o){return dot(o, l.a, l.b);}
inline DB dot(const Line &l1, const Line &l2){return dot(l1.b - l1.a, l2.b - l2.a);}

inline DB det(const DB &x1, const DB &y1, const DB &x2, const DB &y2){return x1 * y2 - x2 * y1;}
inline DB det(const Po &a, const Po &b){return det(a.x, a.y, b.x, b.y);}
inline DB det(const Po &p0, const Po &p1, const Po &p2){return det(p1 - p0, p2 - p0);}
inline DB det(const Po &o, const Line &l){return det(o, l.a, l.b);}
inline DB det(const Line &l, const Po &o){return det(o, l.a, l.b);}
inline DB det(const Line &l1, const Line &l2){return det(l1.b - l1.a, l2.b - l2.a);}

DB Po::operator *(const Po &r)const{return dot(*this, r);}
DB Po::operator ^(const Po &r)const{return det(*this, r);}

void Line::getequation(DB A, DB B, DB C) const{
A = a.y - b.y, B = b.x - a.x, C = det(a, b);
}

template<class T1, class T2> inline DB dist(const T1 &x, const T2 &y){return sqrt(dist_sqr(x, y));}
template<class T1, class T2, class T3> inline DB dist(const T1 &x, const T2 &y, const T3 &z){return sqrt(dist_sqr(x, y, z));}
template<class T1, class T2> inline int dett(const T1 &x, const T2 &y){return sgn(det(x, y));}
template<class T1, class T2> inline int dott(const T1 &x, const T2 &y){return sgn(dot(x, y));}
template<class T1, class T2, class T3> inline int dett(const T1 &x, const T2 &y, const T3 &z){return sgn(det(x, y, z));}
template<class T1, class T2, class T3> inline int dott(const T1 &x, const T2 &y, const T3 &z){return sgn(dot(x, y, z));}
template<class T1, class T2, class T3, class T4> inline int dett(const T1 &x, const T2 &y, const T3 &z, const T4 &w){return sgn(det(x, y, z, w));}
template<class T1, class T2, class T3, class T4> inline int dott(const T1 &x, const T2 &y, const T3 &z, const T4 &w){return sgn(dot(x, y, z, w));}

inline DB dist_sqr(const DB &x, const DB &y){return sqr(x) + sqr(y);}
inline DB dist_sqr(const DB &x, const DB &y, const DB &z){return sqr(x) + sqr(y) + sqr(z);}
inline DB dist_sqr(const Po &a, const Po &b){return sqr(a.x - b.x) + sqr(a.y - b.y);}
inline DB dist_sqr(const Po &p, const Line &l){Po v0 = l.b - l.a, v1 = p - l.a; return sqr(fabs(det(v0, v1))) / v0.length_sqr();}
inline DB dist_sqr(const Po &p, const Seg &l){
Po v0 = l.b - l.a, v1 = p - l.a, v2 = p - l.b;
if (sgn(dot(v0, v1)) * sgn(dot(v0, v2)) <= 0) return dist_sqr(p, Line(l));
else return min(v1.length_sqr(), v2.length_sqr());
}
inline DB dist_sqr(Line l, Po p){return dist_sqr(p, l);}
inline DB dist_sqr(Seg l, Po p){return dist_sqr(p, l);}
inline DB dist_sqr(Line l1, Line l2){
if (sgn(det(l1, l2)) != 0) return 0;
return dist_sqr(l1.a, l2);
}
inline DB dist_sqr(Line l1, Seg l2){
Po v0 = l1.b - l1.a, v1 = l2.a - l1.a, v2 = l2.b - l1.a; DB c1 = det(v0, v1), c2 = det(v0, v2);
return sgn(c1) != sgn(c2) ? 0 : sqr(min(fabs(c1), fabs(c2))) / v0.length_sqr();
}

} using namespace CG;//}
//}

// <<= 'Random Event .. . //{
namespace RNG{
//srand((unsigned)time(NULL));
inline unsigned int rand16(){return (bool(rand()&1) << 15) | rand();}
inline unsigned int rand32(){return (rand16() << 16) | rand16();}
inline ULL rand64(){return ((LL)rand32() << 32) | rand32();}
inline ULL random(LL l, LL r){return rand64() % (r - l) + l;}
int dice(){return rand() % 6;}
bool coin(){return bool(rand() % 2);}
} using namespace RNG;
//}
// <<= 'Clock .. . //{
namespace CLOCK{
DB s0, s1, rd, k, T;
inline DB getTime(){
#ifdef LOCAL
return 1.0 * clock() / CLOCKS_PER_SEC;
#else
timeval tv;
gettimeofday(&tv, 0);
return tv.tv_sec + tv.tv_usec * 1e-6;
#endif
}

inline void st0(DB _T = 0.98){T = _T, s0 = getTime();}
inline void st1(DB _k = 1.618){k = _k, s1 = getTime();}
inline void ed1(){rd = getTime() - s1;}
inline DB elapsed(){return getTime() - s0;}
inline bool safe(){return elapsed() + rd * k < T;}
} //using namespace CLOCK;
//}
// <<= 'Temp .. . //{
namespace TMP{
template<class T> PTT operator+(const PTT &p1, const PTT &p2) {
return PTT(p1.fi + p2.fi, p1.se + p2.se);
}

template<class T> PTT operator-(const PTT &p1, const PTT &p2) {
return PTT(p1.fi - p2.fi, p1.se - p2.se);
}

template<class T> PTT operator*(const PTT &lhs, T k){
return PTT(lhs.fi * k, lhs.se * k);
}
} using namespace TMP;
//}

//}

/** I/O Accelerator Interface .. **/ //{
template<class T> inline T& RD(T &x){
//cin >> x;
//scanf("%d", &x);
char c; for (c = getchar(); c < '0'; c = getchar()); x = c - '0'; for (c = getchar(); '0' <= c ; c = getchar()) x = ((x<<3)+(x<<1)) + c - '0';
//char c; c = getchar(); x = c - '0'; for (c = getchar(); c >= '0'; c = getchar()) x = x * 10 + c - '0'; // && c <= '9'
return x;
}

inline char* RS(char *s){
//gets(s);
scanf("%s", s);
return s;
}

int Case; template<class T> inline void OT(const T &x){
//printf("Case %d: %d\n", ++Case, x);
//printf("%I64d\n", x);
//printf("%.2lf\n", x);
printf("%d\n", x);
//cout << x << endl;
}
//}

//}/* .................................................................................................................................. */

struct Po3D: Po{
DB z;
Po3D(DB x=0, DB y=0, DB z=0):Po(x, y),z(z){}

friend istream& operator >>(istream& in, Po3D &p){return in >> p.x >> p.y >> p.z;}
friend ostream& operator <<(ostream& out, Po3D p){return out << "(" << p.x << ", " << p.y << ", " << p.z << ")";}
bool operator ==(const Po3D& r)const{return !sgn(x-r.x) && !sgn(y-r.y) && !sgn(z-r.z);};
bool operator !=(const Po3D& r)const{return sgn(x-r.x) || sgn(y-r.y) && sgn(z-r.z);}
Po3D operator +(const Po3D& r)const{return Po3D(x+r.x, y+r.y, z+r.z);}
Po3D operator -(const Po3D& r)const{return Po3D(x-r.x, y-r.y, z-r.z);}
Po3D operator *(DB k)const{return Po3D(x*k,y*k,z*k);}
Po3D operator /(DB k)const{return Po3D(x/k,y/k,z/k);}
DB operator *(const Po3D&)const;
Po3D operator ^(const Po3D&)const;

bool operator <(const Po3D &r) const{return sgn(x, r.x)<0 || !sgn(x, r.x) && (sgn(y, r.y)<0 || !sgn(y, r.y) && sgn(z, r.z)<0);}
Po3D operator -()const{return Po3D(-x,-y,-z);}
Po3D& operator +=(const Po3D &r){x+=r.x,y+=r.y,z+=r.z;return*this;}
Po3D& operator -=(const Po3D &r){x-=r.x,y-=r.y,z-=r.z;return*this;}
Po3D& operator *=(DB k){x*=k,y*=k,z*=k;return*this;}
Po3D& operator /=(DB k){x/=k,y/=k,z/=k;return*this;}

DB length_sqr()const{return sqr(x) + sqr(y);}
DB length()const{return sqrt(length_sqr());}
Po unit()const{return (*this) / length();}
bool dgt()const{return !sgn(x) && !sgn(y);}
DB atan()const{return atan2(y, x);}
void input(){RF(x,y,z);}
};

inline DB dot(DB x1, DB y1, DB z1, DB x2, DB y2, DB z2){return x1*x2+y1*y2+z1*z2;}
inline DB dot(const Po3D &a, const Po3D &b){return dot(a.x,a.y,a.z,b.x,b.y, b.z);}
inline DB dot(const Po3D &p0, const Po3D &p1, const Po3D &p2){return dot(p1-p0, p2-p0);}

inline Po3D det(DB x1, DB y1, DB z1, DB x2, DB y2, DB z2){return Po3D(det(y1,z1,y2,z2),det(z1,x1,z2,x2),det(x1,y1,x2,y2));}
inline Po3D det(const Po3D &a, const Po3D &b){return det(a.x,a.y,a.z,b.x,b.y,b.z);}
inline Po3D det(const Po3D &p0, const Po3D &p1, const Po3D &p2){return det(p1-p0, p2-p0);}

inline DB box(const Po3D &a, const Po3D &b, const Po3D& c){return a*(b^c);}
inline DB box(const Po3D &p0, const Po3D &p1, const Po3D& p2, const Po3D& p3){return box(p1-p0,p2-p0,p3-p0);}

DB Po3D::operator *(const Po3D &r)const{return dot(*this, r);}
Po3D Po3D::operator ^(const Po3D &r)const{return det(*this, r);}

const int N = 509;

struct CH3D{

struct fac{
int a, b, c;
bool ok;
} F[N*8];

int to[N][N]; Po3D P[N];
int n, nn;

double ptof(const Po3D &p, fac& f) {
return box(P[f.a], P[f.b], P[f.c], p);
}

void add_face(int a, int b, int c){
F[nn].a = a, F[nn].b = b, F[nn].c = c, F[nn].ok = 1;
to[a][b] = to[b][c] = to[c][a] = nn++;
}

void deal(int p, int a, int b) {
int f = to[a][b]; if (F[f].ok){
if (ptof(P[p], F[f]) > EPS) dfs(p, f);
}
}

void dfs(int p, int cur) {
F[cur].ok = 0;
deal(p, F[cur].b, F[cur].a);
deal(p, F[cur].c, F[cur].b);
deal(p, F[cur].a, F[cur].c);
}

bool same(int s, int t) {
Po3D &a = P[F[s].a], &b = P[F[s].b], &c = P[F[s].c];
return !sgn(box(a, b, c, P[F[t].a])) && !sgn(box(a, b, c, P[F[t].b])) && !sgn(box(a, b, c, P[F[t].c]));
}

void construct() {

assert(n >= 4);

nn = 0; int i;

FOR_N(i, 1, n) if (!(P[0] - P[i]).dgt()){
swap(P[1], P[i]);
break;
}

assert(i < n);

FOR_N(i, 2, n) if (!((P[0]-P[1])^(P[i]-P[1])).dgt()){
swap(P[2], P[i]);
break;
}

assert(i < n);

FOR_N(i, 3, n) if (sgn(box(P[0]-P[1],P[1]-P[2],P[0]-P[i]))) {
swap(P[3], P[i]);
break;
}

assert(i < n);

REP(i, 4){
add.a = (i + 1) % 4, add.b = (i + 2) % 4, add.c = (i + 3) % 4, add.ok = 1;
}

FOR(i, 4, n){
REP(j, nn) if (F[j].ok && ptof(P[i], F[j]) > EPS){
dfs(i, j);
break;
}
}

int t = nn; nn = 0; REP(i, t) if (F[i].ok) F[nn++] = F[i];
}

int face(){
int res = 0; REP(i, nn){
int j; REP_N(j, i) if (same(i, j)) break;
res += j == i;
}
return res;
}

void init(int _n){
n = _n; REP(i, n) P[i].input();
construct();
}
} C;

int n;

int main(){

#ifndef ONLINE_JUDGE
freopen("in.txt", "r", stdin);
#endif

while (scanf("%d", &n) != EOF){
C.init(n), OT(C.face());
}
}

1. 昨晚喝了点小酒，喝高了有点醉，晃晃悠悠的回家，踹开门突然看到老婆和隔壁老王在床上抱在一起睡觉，我当时就气炸肺了，直接过去揪老王起来，给老王脸上上去就打了三拳，然后抓着我老婆扇了三个大嘴巴子，把这两个狗男女给一顿揍，真解气

2. 昨晚喝了点小酒，喝高了有点醉，晃晃悠悠的回家，踹开门突然看到老婆和隔壁老王在床上抱在一起睡觉，我当时就气炸肺了，直接过去揪老王起来，给老王脸上上去就打了三拳，然后抓着我老婆扇了三个大嘴巴子，把这两个狗男女给一顿揍，真解气

3. 昨晚喝了点小酒，喝高了有点醉，晃晃悠悠的回家，踹开门突然看到老婆和隔壁老王在床上抱在一起睡觉，我当时就气炸肺了，直接过去揪老王起来，给老王脸上上去就打了三拳，然后抓着我老婆扇了三个大嘴巴子，把这两个狗男女给一顿揍，真解气

4. 昨晚喝了点小酒，喝高了有点醉，晃晃悠悠的回家，踹开门突然看到老婆和隔壁老王在床上抱在一起睡觉，我当时就气炸肺了，直接过去揪老王起来，给老王脸上上去就打了三拳，然后抓着我老婆扇了三个大嘴巴子，把这两个狗男女给一顿揍，真解气

5. A猴子认识的所有猴子和B猴子认识的所有猴子都能认识，这句话用《爱屋及乌》描述比较容易理解……

6. for(int i=1; i<=m; i++){
for(int j=1; j<=n; j++){
dp = dp [j-1] + 1;
if(s1.charAt(i-1) == s3.charAt(i+j-1))
dp = dp[i-1] + 1;
if(s2.charAt(j-1) == s3.charAt(i+j-1))
dp = Math.max(dp [j - 1] + 1, dp );
}
}
这里的代码似乎有点问题？ dp(i)(j) = dp(i)(j-1) + 1;这个例子System.out.println(ils.isInterleave("aa","dbbca", "aadbbcb"));返回的应该是false

7. 算法是程序的灵魂，算法分简单和复杂，如果不搞大数据类，程序员了解一下简单点的算法也是可以的，但是会算法的一定要会编程才行，程序员不一定要会算法，利于自己项目需要的可以简单了解。