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UVa 10396 Vampire Numbers

news 2025/11/2 11:01:10

题目分析

吸血鬼数（ $Number\texttt{Vampire Number}$ ）是指满足以下条件的数字 $v$ ：

$v$ 有偶数 $n$ 位数字
$\times y$ ，其中 $x$ 和 $y$ 都是 $n /2$ 位的数字
$x$ 和 $y$ 的数字合起来正好是 $v$ 的数字（顺序可以任意）
不允许 $x$ 和 $y$ 同时以 $0$ 结尾
本题要求找出所有偶数的 $n$ 位吸血鬼数（ $n = 4, 6, 8$ ）

解题思路

方法选择

由于题目只需要处理 $4$ 、 $6$ 、 $8$ 位的情况，可以采用预计算 + 直接输出的策略：

预计算所有可能的吸血鬼数：对于每个 $n$ ，遍历所有可能的 $n /2$ 位因子对 $(x, y)$
验证条件：
- 检查乘积 $\times y$ 是否为 $n$ 位数
- 检查 $v$ 是否为偶数
- 检查数字组成是否匹配（通过排序后比较）
- 排除双 $0$ 结尾的情况
缓存结果：对每个 $n$ 只计算一次，后续直接输出

算法优化

范围剪枝：根据 $n$ 位数范围计算 $y$ 的有效取值范围
避免重复：只考虑 $\leq y$ 的情况
提前终止：先检查奇偶性和位数，再进行昂贵的字符串操作

参考代码

// Vampire Numbers
// UVa ID: 10396
// Verdict: Accepted
// Submission Date: 2025-11-01
// UVa Run Time: 1.380s
//
// 版权所有（C）2025，邱秋。metaphysis # yeah dot net#include <bits/stdc++.h>using namespace std;// 使用位运算记录数字频率（每个数字 0-9 用 4 位存储，共 40 位）
long long getDigitSignature(int num) {long long signature = 0;while (num > 0) {int digit = num % 10;signature += (1LL << (digit * 4));  // 每个数字用 4 位计数num /= 10;}return signature;
}// 检查是否为吸血鬼数（纯数学方法）
bool isVampireMath(int x, int y, int v, int n) {// 检查双0结尾if (x % 10 == 0 && y % 10 == 0) return false;// 检查数字组成是否相同long long sig_v = getDigitSignature(v);long long sig_xy = getDigitSignature(x * (int)pow(10, n/2) + y);return sig_v == sig_xy;
}vector<int> findEvenVampireNumbers(int n) {set<int> results;int half = n / 2;int start = pow(10, half - 1);int end = pow(10, half) - 1;// 预计算范围int min_v = pow(10, n - 1);int max_v = pow(10, n) - 1;for (int x = start; x <= end; x++) {// 计算y的有效范围int min_y = max(x, (min_v + x - 1) / x);  // 向上取整int max_y = min(end, max_v / x);for (int y = min_y; y <= max_y; y++) {int v = x * y;// 快速检查：必须是偶数if ((v & 1) != 0) continue;// 使用数学方法检查数字组成if (isVampireMath(x, y, v, n)) {results.insert(v);}}}return vector<int>(results.begin(), results.end());
}// 预计算并缓存结果
vector<int> getPrecomputedEvenVampires(int n) {static vector<int> cache4, cache6, cache8;static bool computed4 = false, computed6 = false, computed8 = false;switch (n) {case 4:if (!computed4) {cache4 = {1260, 1530, 6880};  // 已知的 4 位偶数吸血鬼数computed4 = true;}return cache4;case 6:if (!computed6) {cache6 = findEvenVampireNumbers(6);computed6 = true;}return cache6;case 8:if (!computed8) {cache8 = findEvenVampireNumbers(8);computed8 = true;}return cache8;default:return vector<int>();}
}int main() {cin.tie(0), cout.tie(0), ios::sync_with_stdio(false);int n;vector<int> inputs;while (cin >> n) inputs.push_back(n);for (size_t i = 0; i < inputs.size(); i++) {vector<int> result = getPrecomputedEvenVampires(inputs[i]);for (int v : result) cout << v << '\n';cout << '\n';}return 0;
}

偷懒做法，打表直接输出。

// Vampire Numbers
// UVa ID: 10396
// Verdict: Accepted
// Submission Date: 2025-11-01
// UVa Run Time: 0.000s
//
// 版权所有（C）2025，邱秋。metaphysis # yeah dot net#include <bits/stdc++.h>
using namespace std;
vector<int> evenVampire4 = {1260, 1530, 6880};
vector<int> evenVampire6 = {102510, 104260, 105210, 105264, 105750, 110758, 115672, 118440, 120600, 123354, 125248, 125460, 125500, 126846, 129640, 131242, 132430, 135828, 136948, 140350, 145314, 146952, 150300, 152608, 153436, 156240, 162976, 163944, 172822, 173250, 174370, 182250, 182650, 186624, 190260, 192150, 201852, 211896, 213466, 215860, 217638, 218488, 226498, 226872, 229648, 233896, 241564, 245182, 251896, 253750, 254740, 260338, 262984, 263074, 284598, 284760, 286416, 296320, 315594, 315900, 319536, 326452, 329346, 329656, 336550, 336960, 338296, 346968, 362992, 365638, 368550, 378400, 378418, 378450, 384912, 392566, 404968, 416650, 416988, 428980, 429664, 447916, 456840, 457600, 458640, 475380, 486720, 498550, 529672, 538650, 559188, 567648, 568750, 629680, 638950, 673920, 679500, 729688, 738468, 769792, 789250, 794088, 809964, 815958, 829696, 939658
};
vector<int> evenVampire8 = {10025010, 10042510, 10052010, 10052064, 10081260, 10102252, 10124352, 10129900, 10133788, 10149750, 10165680, 10166350, 10173820, 10185934, 10192248, 10195794, 10205100, 10205370, 10219680, 10233054, 10237864, 10245550, 10245820, 10250460, 10254100, 10255000, 10257790, 10269504, 10269598, 10293966, 10294200, 10311966, 10391746, 10396800, 10402600, 10423530, 10434226, 10437678, 10461784, 10462090, 10467792, 10471162, 10502100, 10507500, 10511694, 10518678, 10522350, 10524208, 10525000, 10526400, 10558368, 10571080, 10572300, 10591398, 10592482, 10642540, 10656432, 10657480, 10665720, 10671660, 10696824, 10699744, 10716300, 10732306, 10732900, 10734192, 10742868, 10767568, 10772190, 10792390, 10796674, 10801350, 10821600, 10832850, 10835370, 10886400, 10905250, 10932754, 10935436, 11053714, 11059164, 11064280, 11085304, 11147836, 11148520, 11165728, 11185762, 11187360, 11246800, 11246890, 11259000, 11272356, 11281648, 11293744, 11344860, 11364138, 11368026, 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};
int main() {cin.tie(0), cout.tie(0), ios::sync_with_stdio(false);int n;vector<int> inputs;while (cin >> n) inputs.push_back(n);for (size_t i = 0; i < inputs.size(); i++) {const vector<int>* result;switch (inputs[i]) {case 4: result = &evenVampire4; break;case 6: result = &evenVampire6; break;case 8: result = &evenVampire8; break;default: break;}for (int v : *result) cout << v << '\n';cout << '\n';}return 0;
}