题目链接:http://acm.hdu.edu.cn/showproblem.php?pid=5312
Sequence
Time Limit: 2000/2000 MS (Java/Others) Memory Limit: 262144/262144 K (Java/Others)
Total Submission(s): 1336 Accepted Submission(s): 410
Problem Description
Today, Soda has learned a sequence whose
n-th(n≥1)
item is 3n(n?1)+1.
Now he wants to know if an integer m
can be represented as the sum of some items of that sequence. If possible, what are the minimum items needed?
For example, 22=19+1+1+1=7+7+7+1.
Input
There are multiple test cases. The first line of input contains an integerT(1≤T≤104),
indicating the number of test cases. For each test case:
There‘s a line containing an integer m(1≤m≤109).
Output
For each test case, output
?1
if m
cannot be represented as the sum of some items of that sequence, otherwise output the minimum items needed.
Sample Input
10 1 2 3 4 5 6 7 8 22 10
Sample Output
1 2 3 4 5 6 1 2 4 4
Source
题目大意:给出一个序列3*n*(n-1)+1。再输入一个m,求构成给定n所需的最小个数。
(序列中的没一个数能够使用若干次)
解题思路:明白一下N*(N-1)/2为三角形数。性质:随意一个自然数都最多可由三个三角形数表示。 题目给的序列是3*n*(n-1)+1就能够转换为6*n*(n-1)/2+1。对于给定的值m,假如m须要k个数来表示。那么其一组解能够表示为m= 6*(K个三角形数的和)+K;
即随意由k个数组成的解 都有 (m-K)%6==0;那么仅仅要找到最小的k即为所求。
此外。对于序列的通式。当n=1或者n=2的时候,就会没有意义,所以对于1和2的时候须要特殊推断一下。
这是一道三角形数的推导及运用题目。
详见代码 。
#include <iostream> #include <cstdio> #include <cstring> #include <cmath> using namespace std; int a[100005]; bool check1(int m) { for (int i=1; i<20010; i++) { if (a[i]==m) return 1; } return 0; } bool check2(int m) { int j; for (int i=1,j=20010-1; i<20010&&a[i]<m; i++) { while (a[i]+a[j]>m) j--; if (a[i]+a[j]==m) return 1; } return 0; } int main() { for (int i=0; i<20010; i++) { a[i]=3*i*(i-1)+1; } int t; scanf("%d",&t); while (t--) { int m; scanf("%d",&m); int flag=0; if (check1(m)) { printf ("1\n"); continue; } else if (check2(m)) { printf ("2\n"); continue; } else { for (int i=3; i<=8; i++) //循环到8的原因是由于模6的余数仅仅有0-5六个 { if ((m-i)%6==0) { printf ("%d\n",i); flag=1; break; } } } if (flag==0) { printf ("-1\n"); } } return 0; }