[Blue Bridge Cup 2015 final] robot breeding
Title Description
Robots in Galaxy X can copy themselves automatically. They can replicate two of themselves in one year, and then lose the ability to replicate.
Every year, Galaxy X selects a new born robot to send to space. That is to say, if there are five robots in Galaxy X,
The total after one year is: 5 + 9 = 14
2 years later: 5 + 9 + 17 = 31
If we have detected the total number of robots s after n years, can you figure out how many robots there were in the first place?
input
Multiple groups of test data exist in the input
For each group of test data, enter two numbers n and s in a row, separated by spaces, meaning as above. n is not more than 100, and s digit is not more than 50.
output
For each group of test data, it is required to output a row, an integer, indicating how many robots there are at first.
sample input
2 31
97 2218388550399401452619230609499
sample output
5
8
Analysis
Assuming that the initial number of robots is a0a, and the number of newly born robots (minus those sent to space) after nnn is ana, then
a0=a0a_0 = a_0a0=a0
a1=2∗a0−1a_1 = 2 * a_0 -1a1=2∗a0−1
a2=2∗a1−1a_2 = 2 * a_1 - 1a2=2∗a1−1
...\quad\quad...
an=2∗an−1−1a_n = 2 * a_{n-1}- 1an=2∗an−1−1
Let a n − t=2 * (an − 1 − T) a_n - t=2 * (a {n-1} - t) an − t=2 * (an − 1 − T)
Then a n = 2 * an − 1 − t a_n = 2 * a {n-1} - Tan = 2 * an − 1 − T, so t=1t = 1t=1
a0=a0a_0 = a_0a0=a0
a1−1=2∗(a0−1)a_1 - 1 = 2 * (a_0 - 1)a1−1=2∗(a0−1)
a2−1=2∗(a1−1)a_2 - 1 = 2 * (a_1 - 1)a2−1=2∗(a1−1)
...\quad\quad ...
an−1=2∗(an−1−1)a_n - 1 = 2 *(a_{n-1} - 1)an−1=2∗(an−1−1)
Taking a1 − 1a_1 - 1a1 − 1 as the first term, 222 is the common ratio, and the sequence of equal ratio is constructed
Then a1 + a2 +... + a n − n=2 * (a0 − 1) × (2n − 1) a 1 + a 2 +... + a n - n=2 * (a 0 - 1) × (2 ^ n - 1) a1+a2+...+an n=2 * (a0 − 1) × (2n − 1)
Then s=2 * (a0 − 1) × (2n − 1) + N + a0s=2 * (a_-1) × (2 ^ n-1) + N + a s=2 * (a0 − 1) × (2n − 1)+n+a0
a0=s+2n+1−2−n2n+1−1a_0 = \frac{s + 2^{n+1} - 2 - n}{2 ^ {n+1} - 1}a0=2n+1−1s+2n+1−2−n
Directly call large number board
# include <iostream> # include <vector> # include <sstream> # include <iomanip> using namespace std; const int base = 1000000000; const int base_digits = 9; struct Int { vector<int> a; int sign; void trim() { while (!a.empty() && !a.back()) a.pop_back(); if (a.empty()) sign = 1; } Int() : sign(1) {}; Int(long long v) { *this = v; } Int(const string &s) { read(s); } int size(){ if(a.empty()) return 0; int ans=(a.size()-1)*base_digits; int ca=a.back(); while(ca) ans++,ca/=10; return ans; } Int operator ^(const Int &v){ Int ans=1,a=*this,b=v; while(!b.isZero()){ if(b%2) ans*=a; a*=a,b/=2; } return ans; } string to_string(){ stringstream ss; ss << *this; string s; ss >> s; return s; } long long longValue() const { long long res = 0; for (int i = a.size() - 1; i >= 0; i--) res = res * base + a[i]; return res * sign; } int sumof(){ string s = to_string(); int ans = 0; for(auto c : s) ans += c - '0'; return ans; } void operator=(const Int &v) { sign = v.sign; a = v.a; } void operator=(long long v) { sign = 1; a.clear(); if (v < 0) sign = -1, v = -v; for (; v > 0; v = v / base) a.push_back(v % base); } Int operator+(const Int &v) const { if (sign == v.sign) { Int res = v; for (int i = 0, carry = 0; i < (int) max(a.size(), v.a.size()) || carry; ++i) { if (i == (int) res.a.size()) res.a.push_back(0); res.a[i] += carry + (i < (int) a.size() ? a[i] : 0); carry = res.a[i] >= base; if (carry) res.a[i] -= base; } return res; } return *this - (-v); } Int operator-(const Int &v) const { if (sign == v.sign) { if (abs() >= v.abs()) { Int res = *this; for (int i = 0, carry = 0; i < (int) v.a.size() || carry; ++i) { res.a[i] -= carry + (i < (int) v.a.size() ? v.a[i] : 0); carry = res.a[i] < 0; if (carry) res.a[i] += base; } res.trim(); return res; } return -(v - *this); } return *this + (-v); } void operator*=(int v) { if (v < 0) sign = -sign, v = -v; for (int i = 0, carry = 0; i < (int) a.size() || carry; ++i) { if (i == (int) a.size()) a.push_back(0); long long cur = a[i] * (long long) v + carry; carry = (int) (cur / base); a[i] = (int) (cur % base); //asm("divl %%ecx" : "=a"(carry), "=d"(a[i]) : "A"(cur), "c"(base)); } trim(); } Int operator*(int v) const { Int res = *this; res *= v; return res; } void operator*=(long long v) { if (v < 0) sign = -sign, v = -v; for (int i = 0, carry = 0; i < (int) a.size() || carry; ++i) { if (i == (int) a.size()) a.push_back(0); long long cur = a[i] * (long long) v + carry; carry = (int) (cur / base); a[i] = (int) (cur % base); //asm("divl %%ecx" : "=a"(carry), "=d"(a[i]) : "A"(cur), "c"(base)); } trim(); } Int operator*(long long v) const { Int res = *this; res *= v; return res; } friend pair<Int, Int> divmod(const Int &a1, const Int &b1) { int norm = base / (b1.a.back() + 1); Int a = a1.abs() * norm; Int b = b1.abs() * norm; Int q, r; q.a.resize(a.a.size()); for (int i = a.a.size() - 1; i >= 0; i--) { r *= base; r += a.a[i]; int s1 = r.a.size() <= b.a.size() ? 0 : r.a[b.a.size()]; int s2 = r.a.size() <= b.a.size() - 1 ? 0 : r.a[b.a.size() - 1]; int d = ((long long) base * s1 + s2) / b.a.back(); r -= b * d; while (r < 0) r += b, --d; q.a[i] = d; } q.sign = a1.sign * b1.sign; r.sign = a1.sign; q.trim(); r.trim(); return make_pair(q, r / norm); } Int operator/(const Int &v) const { return divmod(*this, v).first; } Int operator%(const Int &v) const { return divmod(*this, v).second; } void operator/=(int v) { if (v < 0) sign = -sign, v = -v; for (int i = (int) a.size() - 1, rem = 0; i >= 0; --i) { long long cur = a[i] + rem * (long long) base; a[i] = (int) (cur / v); rem = (int) (cur % v); } trim(); } Int operator/(int v) const { Int res = *this; res /= v; return res; } int operator%(int v) const { if (v < 0) v = -v; int m = 0; for (int i = a.size() - 1; i >= 0; --i) m = (a[i] + m * (long long) base) % v; return m * sign; } void operator+=(const Int &v) { *this = *this + v; } void operator-=(const Int &v) { *this = *this - v; } void operator*=(const Int &v) { *this = *this * v; } void operator/=(const Int &v) { *this = *this / v; } Int operator ++(){ *this += 1; return *this; } Int operator ++(int){ Int temp = *this; *this += 1; return temp; } Int operator --(){ *this -= 1; return *this; } Int operator --(int){ Int temp = *this; *this -= 1; return temp; } bool operator<(const Int &v) const { if (sign != v.sign) return sign < v.sign; if (a.size() != v.a.size()) return a.size() * sign < v.a.size() * v.sign; for (int i = a.size() - 1; i >= 0; i--) if (a[i] != v.a[i]) return a[i] * sign < v.a[i] * sign; return false; } bool operator>(const Int &v) const { return v < *this; } bool operator<=(const Int &v) const { return !(v < *this); } bool operator>=(const Int &v) const { return !(*this < v); } bool operator==(const Int &v) const { return !(*this < v) && !(v < *this); } bool operator!=(const Int &v) const { return *this < v || v < *this; } bool isZero() const { return a.empty() || (a.size() == 1 && !a[0]); } Int operator-() const { Int res = *this; res.sign = -sign; return res; } Int abs() const { Int res = *this; res.sign *= res.sign; return res; } friend Int gcd(const Int &a, const Int &b) { return b.isZero() ? a : gcd(b, a % b); } friend Int lcm(const Int &a, const Int &b) { return a / gcd(a, b) * b; } void read(const string &s) { sign = 1; a.clear(); int pos = 0; while (pos < (int) s.size() && (s[pos] == '-' || s[pos] == '+')) { if (s[pos] == '-') sign = -sign; ++pos; } for (int i = s.size() - 1; i >= pos; i -= base_digits) { int x = 0; for (int j = max(pos, i - base_digits + 1); j <= i; j++) x = x * 10 + s[j] - '0'; a.push_back(x); } trim(); } friend istream& operator>>(istream &stream, Int &v) { string s; stream >> s; v.read(s); return stream; } friend ostream& operator<<(ostream &stream, const Int &v) { if (v.sign == -1) stream << '-'; stream << (v.a.empty() ? 0 : v.a.back()); for (int i = (int) v.a.size() - 2; i >= 0; --i) stream << setw(base_digits) << setfill('0') << v.a[i]; return stream; } static vector<int> convert_base(const vector<int> &a, int old_digits, int new_digits) { vector<long long> p(max(old_digits, new_digits) + 1); p[0] = 1; for (int i = 1; i < (int) p.size(); i++) p[i] = p[i - 1] * 10; vector<int> res; long long cur = 0; int cur_digits = 0; for (int i = 0; i < (int) a.size(); i++) { cur += a[i] * p[cur_digits]; cur_digits += old_digits; while (cur_digits >= new_digits) { res.push_back(int(cur % p[new_digits])); cur /= p[new_digits]; cur_digits -= new_digits; } } res.push_back((int) cur); while (!res.empty() && !res.back()) res.pop_back(); return res; } typedef vector<long long> vll; static vll karatsubaMultiply(const vll &a, const vll &b) { int n = a.size(); vll res(n + n); if (n <= 32) { for (int i = 0; i < n; i++) for (int j = 0; j < n; j++) res[i + j] += a[i] * b[j]; return res; } int k = n >> 1; vll a1(a.begin(), a.begin() + k); vll a2(a.begin() + k, a.end()); vll b1(b.begin(), b.begin() + k); vll b2(b.begin() + k, b.end()); vll a1b1 = karatsubaMultiply(a1, b1); vll a2b2 = karatsubaMultiply(a2, b2); for (int i = 0; i < k; i++) a2[i] += a1[i]; for (int i = 0; i < k; i++) b2[i] += b1[i]; vll r = karatsubaMultiply(a2, b2); for (int i = 0; i < (int) a1b1.size(); i++) r[i] -= a1b1[i]; for (int i = 0; i < (int) a2b2.size(); i++) r[i] -= a2b2[i]; for (int i = 0; i < (int) r.size(); i++) res[i + k] += r[i]; for (int i = 0; i < (int) a1b1.size(); i++) res[i] += a1b1[i]; for (int i = 0; i < (int) a2b2.size(); i++) res[i + n] += a2b2[i]; return res; } Int operator*(const Int &v) const { vector<int> a6 = convert_base(this->a, base_digits, 6); vector<int> b6 = convert_base(v.a, base_digits, 6); vll a(a6.begin(), a6.end()); vll b(b6.begin(), b6.end()); while (a.size() < b.size()) a.push_back(0); while (b.size() < a.size()) b.push_back(0); while (a.size() & (a.size() - 1)) a.push_back(0), b.push_back(0); vll c = karatsubaMultiply(a, b); Int res; res.sign = sign * v.sign; for (int i = 0, carry = 0; i < (int) c.size(); i++) { long long cur = c[i] + carry; res.a.push_back((int) (cur % 1000000)); carry = (int) (cur / 1000000); } res.a = convert_base(res.a, 6, base_digits); res.trim(); return res; } friend Int max(const Int &a,const Int &b){ if(a<b){ return a; } return b; } friend Int max(const Int &a,const int32_t &B){ Int b = B; return max(a,b); } friend Int max(const Int &a,const int64_t &B){ Int b = B; return max(a,b); } friend Int min(const Int &a,const Int &b){ if(a>b){ return b; } return a; } friend Int min(const Int &a,const int32_t &B){ Int b = B; return min(a,b); } friend Int min(const Int &a,const int64_t &B){ Int b = B; return min(a,b); } friend Int pow(const Int &a,const Int &b){ Int _c = 1; Int _b = b; Int _a = a; while(_b != 0){ if(_b%2){ _c *= _a; } _a *= _a; _b /= 2; } return _c; } friend Int pow(const Int &a,const int32_t &B){ Int b = B; return pow(a,b); } friend Int pow(const Int &a,const int64_t &B){ Int b = B; return pow(a,b); } friend Int sqrt(Int a) { Int x0 = a, x1 = (a+1)/2; while (x1 < x0) { x0 = x1; x1 = (x1+a/x1)/2; } return x0; } }; int main(){ Int n; Int s; while (std::cin >> n >> s) { Int t = (s + pow(2, n+1) - 2 - n) / (pow(2, n+1) - 1); std::cout << t << '\n'; } return 0; }
