第一版

README.md

@@ -1,2 +1,11 @@
 # ReciprocalSearch
 在1到100中找10个不同的数，使其倒数和为1。从小到大排列这10个数，有一个不同就视为不同的解。求所有的解。
+
+程序的原始版本不是我写的。
+
+基本思路是用抽屉原则，确定每重循环/递归的搜索上限，进行剪枝。
+比如：最开始的和是1，这个和需要分给10个数，那么至少有一个数分到的值是不小于1/10的，那么第一重循环的上限不大于10。
+
+计算出来后，用大数运算库GMP校验一下，有71312个是对的（未去重）。校验的时候，第10个整数分别用(int)i10、(int)i10 + 1、(int)i10 - 1去尝试。
+
+整个代码也可以不用double来运算，而是用GMP的自然数库来运算。

ReciprocalSearch.cpp

@@ -0,0 +1,122 @@
+#include <stdio.h>
+#include <math.h>
+
+#pragma warning(disable:4146)
+#include <gmp.h>
+
+#define min(a,b) (((a)<(b))?(a):(b))
+
+bool check(int i1, int i2, int i3, int i4, int i5, int i6, int i7, int i8, int i9, int i10) {
+    int a[10];
+
+    a[0] = i1;
+    a[1] = i2;
+    a[2] = i3;
+    a[3] = i4;
+    a[4] = i5;
+    a[5] = i6;
+    a[6] = i7;
+    a[7] = i8;
+    a[8] = i9;
+    a[9] = i10;
+
+    mpz_t t;
+    mpz_init(t);
+    mpz_set_ui(t, 1);
+    for (int k = 0; k < 10; ++k) {
+        mpz_mul_ui(t, t, a[k]);
+    }
+
+    mpz_t s;
+    mpz_init(s);
+    mpz_set_ui(s, 0);
+
+    mpz_t t1;
+    mpz_init(t1);
+    for (int k = 0; k < 10; ++k) {
+        mpz_div_ui(t1, t, a[k]);
+        mpz_add(s, s, t1);
+    }
+
+    bool valid = (mpz_cmp(s, t) == 0);
+
+    mpz_clear(t);
+    mpz_clear(s);
+    mpz_clear(t1);
+
+    return valid;
+}
+
+int main() {
+    int total = 0;
+    int valid = 0;
+
+    double ii[101];
+
+    for (int i = 1; i <= 100; i++) { ii[i] = 1.0 / i; }
+
+    for (int i1 = 2; i1 <= 10; i1++) {
+        double m1 = 1 - ii[i1];
+
+        for (int i2 = i1 + 1; i2 < min(92, (int)(9.0 / m1) + 1); i2++) {
+            double m2 = m1 - ii[i2];
+
+            for (int i3 = i2 + 1; i3 < min(93, (int)(8.0 / m2) + 1); i3++) {
+                double m3 = m2 - ii[i3];
+                if (m3 <= 0) { continue; }
+
+                for (int i4 = i3 + 1; i4 < min(94, (int)(7.0 / m3) + 1); i4++) {
+                    double m4 = m3 - ii[i4];
+                    if (m4 <= 0) { continue; }
+
+                    for (int i5 = i4 + 1; i5 < min(95, (int)(6.0 / m4) + 1); i5++) {
+                        double m5 = m4 - ii[i5];
+                        if (m5 <= 0) { continue; }
+
+                        for (int i6 = i5 + 1; i6 < min(96, (int)(5.0 / m5) + 1); i6++) {
+                            double m6 = m5 - ii[i6];
+                            if (m6 <= 0) { continue; }
+
+                            for (int i7 = i6 + 1; i7 < min(97, (int)(4.0 / m6) + 1); i7++) {
+                                double m7 = m6 - ii[i7];
+                                if (m7 <= 0) { continue; }
+
+                                for (int i8 = i7 + 1; i8 < min(98, (int)(3.0 / m7) + 1); i8++) {
+                                    double m8 = m7 - ii[i8];
+                                    if (m8 <= 0) { continue; }
+
+                                    for (int i9 = i8 + 1; i9 < min(99, (int)(2.0 / m8) + 1); i9++) {
+                                        double m9 = m8 - ii[i9];
+
+                                        if (m9 <= 0.01) { continue; }
+                                        double i10 = 1.0 / m9;
+
+                                        if (fabs(i10 - round(i10)) < 0.000001) {
+                                            ++total;                                            
+
+                                            if (check(i1, i2, i3, i4, i5, i6, i7, i8, i9, (int)i10)) {
+                                                ++valid;
+                                                printf("%d %d %d %d %d %d %d %d %d %.4f(%d)\n", i1, i2, i3, i4, i5, i6, i7, i8, i9, i10, (int)i10);
+                                            } else if (check(i1, i2, i3, i4, i5, i6, i7, i8, i9, (int)i10 + 1)) {
+                                                ++valid;
+                                                printf("%d %d %d %d %d %d %d %d %d %.4f(%d)\n", i1, i2, i3, i4, i5, i6, i7, i8, i9, i10, (int)i10 + 1);
+                                            } else if (check(i1, i2, i3, i4, i5, i6, i7, i8, i9, (int)i10 - 1)) {
+                                                ++valid;
+                                                printf("%d %d %d %d %d %d %d %d %d %.4f(%d)\n", i1, i2, i3, i4, i5, i6, i7, i8, i9, i10, (int)i10 - 1);
+                                            }
+                                            else {
+                                                printf("%d %d %d %d %d %d %d %d %d %.4f    INVALID\n", i1, i2, i3, i4, i5, i6, i7, i8, i9, i10);
+                                            }
+                                        }
+                                    }
+                                }
+                            }
+                        }
+                    }
+                }
+            }
+        }
+    }
+
+    printf("total %d, valid = %d\n", total, valid);
+}