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## Project Euler 45–Triangular, pentagonal, and hexagonal

Wu Yudong    August 20, 2018     欧拉计划   618

Triangular, pentagonal, and hexagonal

Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:

 Triangle Tn=n(n+1)/2 1, 3, 6, 10, 15, … Pentagonal Pn=n(3n−1)/2 1, 5, 12, 22, 35, … Hexagonal Hn=n(2n−1) 1, 6, 15, 28, 45, …

It can be verified that T285 = P165 = H143 = 40755.

Find the next triangle number that is also pentagonal and hexagonal.

 三角形数 Tn=n(n+1)/2 1, 3, 6, 10, 15, … 五边形数 Pn=n(3n−1)/2 1, 5, 12, 22, 35, … 六边形数 Hn=n(2n−1) 1, 6, 15, 28, 45, …

$$\frac{n(n+1)}{2}=m$$

$$n=\frac{-1+\sqrt{1+8m}}{2}$$

$$\frac{n(3n-1)}{2}=m$$

$$n=\frac{1+\sqrt{1+24m}}{6}$$

$$n(2n-1)=m$$

$$n=\frac{1+\sqrt{1+8m}}{4}$$

// Project Euler 45–Triangular, pentagonal, and hexagonal
// Completed on Mon, 20 Aug 2018, 23:12
// Language: C
//
// 版权所有（C）wu yudong
// 博客地址：http://www.wuyudong.com

#include<stdio.h>
#include<math.h>
#include<string.h>
#include<ctype.h>
#include<stdlib.h>
#include<stdbool.h>
#include<time.h>

int main()
{
long long ans;
int begintime, endtime;
int n = 143;
begintime = clock();		//计时开始
while (1) {
n++;
//Hexagonal
double h = n * (2 * n - 1);
double Triangle = (sqrt(1 + 8 * h) - 1) / 2;
double Pentagonal = (sqrt(1 + 24 * h) + 1) / 6;
double T = (int)Triangle;
double P = (int)Pentagonal;

if (T == Triangle && P == Pentagonal) {
ans = n * (2 * n - 1);
printf("%lld\n", ans);
break;
}
}
endtime = clock();			//计时结束

printf("Running Time: %dms\n", endtime - begintime);
return 0;
}


1533776805
Running Time: 1ms

Completed on Mon, 20 Aug 2018, 23:12