题解 - [Luogu P2277] [HNOI2003] 24 点游戏

题目链接

原始题面

题目描述

为了培养小孩的计算能力,大人们经常给小孩玩这样的游戏:从 1 付扑克牌中任意抽出 4 张扑克,要小孩用 "+", "-", "*", "/" 和括号组成一个合法的表达式,并使表达式的值为 24 点。这种游戏就是所谓的 "24 点游戏"

请你编程求出对于给出的任意 4 个正整数 a, b, c, d, 请你编程求出这 4 个整数能组成多少个值为 24 的不同表达式

输入输出格式

输入格式

输入文件名: input.txt

输入文件共一行,为 4 个正整数 a, b, c, d. (0<=a,b,c,d<=100)

输出格式

输出文件名: output.txt

输出文件为由 a, b, c, d 组成的值为 24 表达式个数,如没有,输出 0

输入输出样例

输入样例 #1

1
5 5 5 5

输出样例 #1

1
1

解题思路

整点 nasty 的花活

显然枚举所有可能的式子然后去重就行

这个代码

当然,这代码都 600+ KB 了,没有一个 OJ 会接受这么大的代码,所以我们可以用宏定义折叠一下 (实际上这个代码就是宏定义展开得到的)

我们思考一下需要哪些宏定义

  1. 我们需要一个能枚举 \(\{a,b,c,d\}^3\) (乘法为 Descartes 积) 的宏定义来枚举所有运算符的情况

    Show code

    然后我们需要这样的宏定义来得到每一组运算符

    Show code

  2. 我们需要一个能枚举 \(\{a,b,c,d\}\) 全排列的宏定义来枚举所有数的情况 (实际上,因为这样会让编译时间过长甚至让编译器爆栈,所以这个并没有用上)

    Show code

    然后我们需要这样的宏定义来得到每一组数

    Show code

  3. 我们需要枚举 5 种运算方式

    • \((((a\odot b)\odot c)\odot d)\)
    • \(((a\odot (b\odot c))\odot d)\)
    • \((a\odot ((b\odot c)\odot d))\)
    • \((a\odot (b\odot (c\odot d)))\)
    • \(((a\odot b)\odot (c\odot d))\)

    其中 \(\odot\in\{+,-,\times,\div\}\)

    这个写成宏定义就很简单

    Show code

接下来就是把这三部分拼接起来就好了

定义个结构体 Node 存 hash 和 运算结果,去重用 std::set<Node> 即可

  • Q1: 为什么变长参数宏定义使用了 C 风格的 __VA_ARGS__

  • A1: 因为这样能实现的功能多

    比如 g++#define EXPAND(args...) ##args 会报错,而 #define EXPAND(...) __VA_ARGS__ 不会

  • Q2: 为什么不把去重和 hash 也宏定义化?

  • A2: 两个原因

另外,由于代码中一定要用到列表初始化来构造容器,所以不能在一次构造中塞太多元素,否则会让编译时间大幅增加甚至让编译器爆栈 从而得到一个没有编译信息的 CE 记录

这也就是为什么我代码里向 std::set 加元素是分了 24 次 merge, 而不是一步到位,直接将 7680 个元素一起塞进去

而且即使是这样,洛谷的编译器也会炸掉,所以只能将全排列枚举部分换成 std::next_permutation

代码参考

Show code

Luogu_P2277view raw
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/*
* @Author: Tifa
* @Description: From <https://github.com/Tiphereth-A/CP-archives>
* !!! ATTENEION: All the context below is licensed under a
* GNU Affero General Public License, Version 3.
* See <https://www.gnu.org/licenses/agpl-3.0.txt>.
*/
#include <bits/stdc++.h>
using namespace std;
#define _SEL3(_1, _2, _3, ...) _1, _2, _3
#define SEL3(...) _SEL3(__VA_ARGS__)
#define _CPROD1_4_HELPER(a, b, c, d, ...) \
a, __VA_ARGS__, b, __VA_ARGS__, c, __VA_ARGS__, d, __VA_ARGS__
#define _CPROD2_4_HELPER(a, b, c, d, ...) \
_CPROD1_4_HELPER(a, b, c, d, a, __VA_ARGS__), \
_CPROD1_4_HELPER(a, b, c, d, b, __VA_ARGS__), \
_CPROD1_4_HELPER(a, b, c, d, c, __VA_ARGS__), \
_CPROD1_4_HELPER(a, b, c, d, d, __VA_ARGS__)
#define CPROD3_4(a, b, c, d) \
_CPROD2_4_HELPER(a, b, c, d, a), _CPROD2_4_HELPER(a, b, c, d, b), \
_CPROD2_4_HELPER(a, b, c, d, c), _CPROD2_4_HELPER(a, b, c, d, d)
#define RMAB(a, b, ...) _RM_##b(RM##a, __VA_ARGS__)
#define OPANDS CPROD3_4(+, -, *, /)
#define OPS0 SEL3(OPANDS)
#define OPS1 SEL3(RMAB(3, 1, OPANDS))
#define OPS2 SEL3(RMAB(3, 2, OPANDS))
#define OPS3 SEL3(RMAB(3, 3, OPANDS))
#define OPS4 SEL3(RMAB(3, 4, OPANDS))
#define OPS5 SEL3(RMAB(3, 5, OPANDS))
#define OPS6 SEL3(RMAB(3, 6, OPANDS))
#define OPS7 SEL3(RMAB(3, 7, OPANDS))
#define OPS8 SEL3(RMAB(3, 8, OPANDS))
#define OPS9 SEL3(RMAB(3, 9, OPANDS))
#define OPS10 SEL3(RMAB(3, 10, OPANDS))
#define OPS11 SEL3(RMAB(3, 11, OPANDS))
#define OPS12 SEL3(RMAB(3, 12, OPANDS))
#define OPS13 SEL3(RMAB(3, 13, OPANDS))
#define OPS14 SEL3(RMAB(3, 14, OPANDS))
#define OPS15 SEL3(RMAB(3, 15, OPANDS))
#define OPS16 SEL3(RMAB(3, 16, OPANDS))
#define OPS17 SEL3(RMAB(3, 17, OPANDS))
#define OPS18 SEL3(RMAB(3, 18, OPANDS))
#define OPS19 SEL3(RMAB(3, 19, OPANDS))
#define OPS20 SEL3(RMAB(3, 20, OPANDS))
#define OPS21 SEL3(RMAB(3, 21, OPANDS))
#define OPS22 SEL3(RMAB(3, 22, OPANDS))
#define OPS23 SEL3(RMAB(3, 23, OPANDS))
#define OPS24 SEL3(RMAB(3, 24, OPANDS))
#define OPS25 SEL3(RMAB(3, 25, OPANDS))
#define OPS26 SEL3(RMAB(3, 26, OPANDS))
#define OPS27 SEL3(RMAB(3, 27, OPANDS))
#define OPS28 SEL3(RMAB(3, 28, OPANDS))
#define OPS29 SEL3(RMAB(3, 29, OPANDS))
#define OPS30 SEL3(RMAB(3, 30, OPANDS))
#define OPS31 SEL3(RMAB(3, 31, OPANDS))
#define OPS32 SEL3(RMAB(3, 32, OPANDS))
#define OPS33 SEL3(RMAB(3, 33, OPANDS))
#define OPS34 SEL3(RMAB(3, 34, OPANDS))
#define OPS35 SEL3(RMAB(3, 35, OPANDS))
#define OPS36 SEL3(RMAB(3, 36, OPANDS))
#define OPS37 SEL3(RMAB(3, 37, OPANDS))
#define OPS38 SEL3(RMAB(3, 38, OPANDS))
#define OPS39 SEL3(RMAB(3, 39, OPANDS))
#define OPS40 SEL3(RMAB(3, 40, OPANDS))
#define OPS41 SEL3(RMAB(3, 41, OPANDS))
#define OPS42 SEL3(RMAB(3, 42, OPANDS))
#define OPS43 SEL3(RMAB(3, 43, OPANDS))
#define OPS44 SEL3(RMAB(3, 44, OPANDS))
#define OPS45 SEL3(RMAB(3, 45, OPANDS))
#define OPS46 SEL3(RMAB(3, 46, OPANDS))
#define OPS47 SEL3(RMAB(3, 47, OPANDS))
#define OPS48 SEL3(RMAB(3, 48, OPANDS))
#define OPS49 SEL3(RMAB(3, 49, OPANDS))
#define OPS50 SEL3(RMAB(3, 50, OPANDS))
#define OPS51 SEL3(RMAB(3, 51, OPANDS))
#define OPS52 SEL3(RMAB(3, 52, OPANDS))
#define OPS53 SEL3(RMAB(3, 53, OPANDS))
#define OPS54 SEL3(RMAB(3, 54, OPANDS))
#define OPS55 SEL3(RMAB(3, 55, OPANDS))
#define OPS56 SEL3(RMAB(3, 56, OPANDS))
#define OPS57 SEL3(RMAB(3, 57, OPANDS))
#define OPS58 SEL3(RMAB(3, 58, OPANDS))
#define OPS59 SEL3(RMAB(3, 59, OPANDS))
#define OPS60 SEL3(RMAB(3, 60, OPANDS))
#define OPS61 SEL3(RMAB(3, 61, OPANDS))
#define OPS62 SEL3(RMAB(3, 62, OPANDS))
#define OPS63 RMAB(3, 63, OPANDS)
#define ___CALC0(a, b, c, d, opab, opbc, opcd) \
Node(0, \
a, \
b, \
c, \
d, \
#opab, \
#opbc, \
#opcd, \
((double(a) opab double(b))opbc double(c))opcd double(d))
#define ___CALC1(a, b, c, d, opab, opbc, opcd) \
Node(1, \
a, \
b, \
c, \
d, \
#opab, \
#opbc, \
#opcd, \
(double(a) opab(double(b) opbc double(c)))opcd double(d))
#define ___CALC2(a, b, c, d, opab, opbc, opcd) \
Node(2, \
a, \
b, \
c, \
d, \
#opab, \
#opbc, \
#opcd, \
double(a) opab((double(b) opbc double(c))opcd double(d)))
#define ___CALC3(a, b, c, d, opab, opbc, opcd) \
Node(3, \
a, \
b, \
c, \
d, \
#opab, \
#opbc, \
#opcd, \
double(a) opab(double(b) opbc(double(c) opcd double(d))))
#define ___CALC4(a, b, c, d, opab, opbc, opcd) \
Node(4, \
a, \
b, \
c, \
d, \
#opab, \
#opbc, \
#opcd, \
((double(a) opab double(b))opbc(double(c) opcd double(d))))
#define ___CALC(a, b, c, d, opab, opbc, opcd) \
___CALC0(a, b, c, d, opab, opbc, opcd), \
___CALC1(a, b, c, d, opab, opbc, opcd), \
___CALC2(a, b, c, d, opab, opbc, opcd), \
___CALC3(a, b, c, d, opab, opbc, opcd), \
___CALC4(a, b, c, d, opab, opbc, opcd)
#define __CALC(...) ___CALC(__VA_ARGS__)
#define _CALC(a, b, c, d) \
__CALC(a, b, c, d, OPS0), __CALC(a, b, c, d, OPS1), \
__CALC(a, b, c, d, OPS2), __CALC(a, b, c, d, OPS3), \
__CALC(a, b, c, d, OPS4), __CALC(a, b, c, d, OPS5), \
__CALC(a, b, c, d, OPS6), __CALC(a, b, c, d, OPS7), \
__CALC(a, b, c, d, OPS8), __CALC(a, b, c, d, OPS9), \
__CALC(a, b, c, d, OPS10), __CALC(a, b, c, d, OPS11), \
__CALC(a, b, c, d, OPS12), __CALC(a, b, c, d, OPS13), \
__CALC(a, b, c, d, OPS14), __CALC(a, b, c, d, OPS15), \
__CALC(a, b, c, d, OPS16), __CALC(a, b, c, d, OPS17), \
__CALC(a, b, c, d, OPS18), __CALC(a, b, c, d, OPS19), \
__CALC(a, b, c, d, OPS20), __CALC(a, b, c, d, OPS21), \
__CALC(a, b, c, d, OPS22), __CALC(a, b, c, d, OPS23), \
__CALC(a, b, c, d, OPS24), __CALC(a, b, c, d, OPS25), \
__CALC(a, b, c, d, OPS26), __CALC(a, b, c, d, OPS27), \
__CALC(a, b, c, d, OPS28), __CALC(a, b, c, d, OPS29), \
__CALC(a, b, c, d, OPS30), __CALC(a, b, c, d, OPS31), \
__CALC(a, b, c, d, OPS32), __CALC(a, b, c, d, OPS33), \
__CALC(a, b, c, d, OPS34), __CALC(a, b, c, d, OPS35), \
__CALC(a, b, c, d, OPS36), __CALC(a, b, c, d, OPS37), \
__CALC(a, b, c, d, OPS38), __CALC(a, b, c, d, OPS39), \
__CALC(a, b, c, d, OPS40), __CALC(a, b, c, d, OPS41), \
__CALC(a, b, c, d, OPS42), __CALC(a, b, c, d, OPS43), \
__CALC(a, b, c, d, OPS44), __CALC(a, b, c, d, OPS45), \
__CALC(a, b, c, d, OPS46), __CALC(a, b, c, d, OPS47), \
__CALC(a, b, c, d, OPS48), __CALC(a, b, c, d, OPS49), \
__CALC(a, b, c, d, OPS50), __CALC(a, b, c, d, OPS51), \
__CALC(a, b, c, d, OPS52), __CALC(a, b, c, d, OPS53), \
__CALC(a, b, c, d, OPS54), __CALC(a, b, c, d, OPS55), \
__CALC(a, b, c, d, OPS56), __CALC(a, b, c, d, OPS57), \
__CALC(a, b, c, d, OPS58), __CALC(a, b, c, d, OPS59), \
__CALC(a, b, c, d, OPS60), __CALC(a, b, c, d, OPS61), \
__CALC(a, b, c, d, OPS62), __CALC(a, b, c, d, OPS63),
#define CALC(...) _CALC(__VA_ARGS__)
int dict[256];
dict['+'] = 0;
dict['-'] = 1;
dict['*'] = 2;
dict['/'] = 3;
return 0;
}
();
struct Node {
u64 hash;
double data;
Node(const int &type,
i64 a,
i64 b,
i64 c,
i64 d,
const char *opab,
const char *opbc,
const char *opcd,
const double &_data)
: data(_data) {
switch (type) {
case 0: break;
case 1:
if (!valid(dict[*opab], dict[*opbc])) {
data = 0;
break;
}
break;
case 2:
if (!valid(dict[*opab], dict[*opcd])) {
data = 0;
break;
}
break;
case 3:
if (!valid(dict[*opab], dict[*opbc]) ||
!valid(dict[*opbc], dict[*opcd])) {
data = 0;
break;
}
break;
case 4:
if (!valid(dict[*opbc], dict[*opcd])) {
data = 0;
break;
}
if (!(dict[*opbc] & 1)) {
} else {
}
break;
}
hash = 0;
(hash += a) <<= 7;
(hash += b) <<= 7;
(hash += c) <<= 7;
(hash += d) <<= 2;
(hash += dict[*opab]) <<= 2;
(hash += dict[*opbc]) <<= 2;
(hash += dict[*opcd]) <<= 3;
hash += type;
#undef minmax
}
constexpr bool valid(const int &a, const int &b) const {
if (a == 0) return b > 1;
if (a == 2) return b < 2;
return true;
}
bool operator<(const Node &rhs) const { return hash < rhs.hash; }
};
void decode(u64 hash) {
#define _decode(val, n) \
val = hash & ((1 << n) - 1); \
hash >>= n
static const char _[] = "+-*/";
int type, a, b, c, d, opab, opbc, opcd;
_decode(type, 3);
_decode(opcd, 2);
_decode(opbc, 2);
_decode(opab, 2);
_decode(d, 7);
_decode(c, 7);
_decode(b, 7);
_decode(a, 7);
switch (type) {
case 0:
cerr << "(((" << a << " " << _[opab] << " " << b << ") " << _[opbc] << " "
<< c << ") " << _[opcd] << " " << d << ")" << endl;
break;
case 1:
cerr << "((" << a << " " << _[opab] << " (" << b << " " << _[opbc] << " "
<< c << ")) " << _[opcd] << " " << d << ")" << endl;
break;
case 2:
cerr << "(" << a << " " << _[opab] << " ((" << b << " " << _[opbc] << " "
<< c << ") " << _[opcd] << " " << d << "))" << endl;
break;
case 3:
cerr << "(" << a << " " << _[opab] << " (" << b << " " << _[opbc] << " ("
<< c << " " << _[opcd] << " " << d << ")))" << endl;
break;
case 4:
cerr << "((" << a << " " << _[opab] << " " << b << ") " << _[opbc] << " ("
<< c << " " << _[opcd] << " " << d << "))" << endl;
break;
}
#undef _decode
}
int main() {
i64 a, b, c, d;
cin >> a >> b >> c >> d;
set<Node> s;
vector<double> v{a, b, c, d};
sort(v.begin(), v.end());
do {
s.merge(decltype(s){CALC(v[0], v[1], v[2], v[3])});
} while (next_permutation(v.begin(), v.end()));
int cnt = 0;
for (const auto &[hash, data] : s)
if (abs(data - 24) <= 1e-7) { ++cnt; }
cout << cnt;
return 0;
}