[POJ 1006] Biorhythms C++解题

浏览数：76 / 时间：2015年06月08日

Biorhythms

Time Limit: 1000MS		Memory Limit: 10000K
Total Submissions: 107569		Accepted: 33365

Description

Some people believe that there are three cycles in a person‘s life that start the day he or she is born. These three cycles are the physical, emotional, and intellectual cycles, and they have periods of lengths 23, 28, and 33 days, respectively. There is one peak in each period of a cycle. At the peak of a cycle, a person performs at his or her best in the corresponding field (physical, emotional or mental). For example, if it is the mental curve, thought processes will be sharper and concentration will be easier.
Since the three cycles have different periods, the peaks of the three cycles generally occur at different times. We would like to determine when a triple peak occurs (the peaks of all three cycles occur in the same day) for any person. For each cycle, you will be given the number of days from the beginning of the current year at which one of its peaks (not necessarily the first) occurs. You will also be given a date expressed as the number of days from the beginning of the current year. You task is to determine the number of days from the given date to the next triple peak. The given date is not counted. For example, if the given date is 10 and the next triple peak occurs on day 12, the answer is 2, not 3. If a triple peak occurs on the given date, you should give the number of days to the next occurrence of a triple peak.

Input

You will be given a number of cases. The input for each case consists of one line of four integers p, e, i, and d. The values p, e, and i are the number of days from the beginning of the current year at which the physical, emotional, and intellectual cycles peak, respectively. The value d is the given date and may be smaller than any of p, e, or i. All values are non-negative and at most 365, and you may assume that a triple peak will occur within 21252 days of the given date. The end of input is indicated by a line in which p = e = i = d = -1.

Output

For each test case, print the case number followed by a message indicating the number of days to the next triple peak, in the form:

Case 1: the next triple peak occurs in 1234 days.

Use the plural form ``days‘‘ even if the answer is 1.

Sample Input

0 0 0 0
0 0 0 100
5 20 34 325
4 5 6 7
283 102 23 320
203 301 203 40
-1 -1 -1 -1

Sample Output

Case 1: the next triple peak occurs in 21252 days.
Case 2: the next triple peak occurs in 21152 days.
Case 3: the next triple peak occurs in 19575 days.
Case 4: the next triple peak occurs in 16994 days.
Case 5: the next triple peak occurs in 8910 days.
Case 6: the next triple peak occurs in 10789 days.

翻译：

生理周期

Time Limit: 1000MS		Memory Limit: 10000K
Total Submissions: 107569		Accepted: 33365

Description

人生来就有三个生理周期，分别为体力、感情和智力周期，它们的周期长度为23天、28天和33天。每一个周期中有一天是高峰。在高峰这天，人会在相应的方面表现出色。例如，智力周期的高峰，人会思维敏捷，精力容易高度集中。因为三个周期的周长不同，所以通常三个周期的高峰不会落在同一天。对于每个人，我们想知道何时三个高峰落在同一天。对于每个周期，我们会给出从当前年份的第一天开始，到出现高峰的天数（不一定是第一次高峰出现的时间）。你的任务是给定一个从当年第一天开始数的天数，输出从给定时间开始（不包括给定时间）下一次三个高峰落在同一天的时间（距给定时间的天数）。例如：给定时间为10，下次出现三个高峰同天的时间是12，则输出2（注意这里不是3）。

Input

输入四个整数：p, e, i和d。 p, e, i分别表示体力、情感和智力高峰出现的时间（时间从当年的第一天开始计算）。d 是给定的时间，可能小于p, e, 或 i。所有给定时间是非负的并且小于365, 所求的时间小于21252。

当p = e = i = d = -1时，输入数据结束。

Output

从给定时间起，下一次三个高峰同天的时间（距离给定时间的天数）。

采用以下格式：
Case 1: the next triple peak occurs in 1234 days.

注意：即使结果是1天，也使用复数形式“days”。

Sample Input

0 0 0 0
0 0 0 100
5 20 34 325
4 5 6 7
283 102 23 320
203 301 203 40
-1 -1 -1 -1

Sample Output

Case 1: the next triple peak occurs in 21252 days.
Case 2: the next triple peak occurs in 21152 days.
Case 3: the next triple peak occurs in 19575 days.
Case 4: the next triple peak occurs in 16994 days.
Case 5: the next triple peak occurs in 8910 days.
Case 6: the next triple peak occurs in 10789 days.

解决思路

中国剩余定理，本题难点不在编程，而是分析题目并转化为数学公式

要引入本题解法，先来看一个故事 “韩信点兵”：
传说西汉大将韩信，由于比较年轻，开始他的部下对他不很佩服。有一次阅兵时，韩信要求士兵分三路纵队，结果末尾多2人，改成五路纵队，结果末尾多3人，再改成七路纵队，结果又余下2人，后来下级军官向他报告共有士兵2395人，韩信立即笑笑说不对（因2395除以3余数是1，不是2），由于已经知道士兵总人数在2300~2400之间，所以韩信根据23，128，233，------，每相邻两数的间隔是105（3、5、7的最小公倍数），便立即说出实际人数应是2333人（因2333=128+20χ105+105，它除以3余2，除以5余3，除以7余2）。这样使下级军官十分敬佩，这就是韩信点兵的故事。

韩信点兵问题简化：已知 n%3=2, n%5=3, n%7=2, 求n。

再看我们这道题，读入p,e,i,d 4个整数

已知(n+d)%23=p; (n+d)%28=e; (n+d)%33=i ,求n 。

两道题是一样的。但是韩信当时计算出结果的？
韩信用的就是“中国剩余定理”，《孙子算经》中早有计算方法，大家可以查阅相关资料。
“韩信点兵”问题计算如下：

因为n%3=2, n%5=3, n%7=2
我们先找一个最小的数能被5和7整除，并且除以3的余数为1，这里，我们可以找到最小的数是70。有什么好处呢，大家知道70 * 1 % 3 = 1，那么70 * 2 % 3 = 2，70 * a % 3 = a,这样可以方便找到140这个除以3余数又为2的数，同时又能被5和7整除。

同理，我们可以找到21是能被3和7整除但是除以5余数是1的数，那么63便是除以5余数是3. 15能被3和5整除，但是除以7余数是1，那么30便是除以7余数是2的数。

那么 140 + 63 + 30 = 233，必定是一个除以3余数是2，除以5余数是3，除以7余数是2的一个数，而我们又很容易知道233加上或者减去3,5,7的最小公倍数，得到的数同样满足这个性质。韩信已知士兵人数在2300~2400之间，所以只需要233 + i × lcm(3,5,7)= 233 + 105 * 20 = 2333.

同样，这道题的解法就是：

假设过n天后，三个生理周期同时到高峰。

已知(n+d)%23=p; (n+d)%28=e; (n+d)%33=i
       使33×28×a被23除余1，用33×28×8=5544；
       使23×33×b被28除余1，用23×33×19=14421；
       使23×28×c被33除余1，用23×28×2=1288。
      因此有（5544×p+14421×e+1288×i）% lcm(23,28,33) =n+d

又23、28、33互质，即lcm(23,28,33)= 21252;
所以有n=（5544×p+14421×e+1288×i-d）%21252

本题所求的是最小整数解，避免n为负，因此最后结果为n= [n+21252]% 21252
那么最终求解n的表达式就是：

n=(5544*p+14421*e+1288*i-d+21252)%21252;

当问题被转化为一条数学式子时，你会发现它无比简单。。。。直接输出结果了。

源码

 1 /*
 2 poj 1000
 3 version:1.0
 4 author:Knight
 5 Email:[email protected]
 6 */
 7 
 8 
 9 #include<iostream>
10 using namespace std;
11  
12 int main(void)
13 {
14     int p,e,i,d;
15     int time=1;
16     while(cin>>p>>e>>i>>d)
17     {
18         if(p==-1 && e==-1 && i==-1 && d==-1)
19             break;
20  
21         int lcm=21252;  // lcm(23,28,33)
22         int n=(5544*p+14421*e+1288*i-d+21252)%21252;
23         if(n==0)
24             n=21252;
25         cout<<"Case "<<time++<<": the next triple peak occurs in "<<n<<" days."<<endl;
26     }
27     return 0;
28 }