100steps

/*
* remember the 100-steps ladder at the university of my graduation.
* it's an unforgotable place.
* these steps in hurry to another classroom,
* sometimes one by one step, sometimes three into two steps and even one step.
* nerver thoungh of the process thought the ladder. How may ways to pass
* throught the amazing place.
* what a pity!
* today is the time to make up for these hurry-missing steps.
*
* 11/19/2012 14:11:40
* */

#include <stdio.h>
#include <string.h>
#define N 100

/*
* result[i] 记录 i 步阶梯的走法数
* */
unsigned long long result[N+1];

/*
* n步阶梯，有三种走法：
* 1、先是跨一步，剩下n-1步，设剩下n-1步，有f(n-1)中走法；
* 2、是跨两步，剩下n-2步，设剩下n-2步，有f(n-2)中走法；
* 3、先是跨三步，剩下n-3步，设剩下n-3步，有f(n-3)中走法；
* n步阶梯总的走法f(n) = f(n-1) + f(n-2) + f(n-3);
*
* 剩下的阶梯，递归这个过程。
* */
unsigned long long howmany_ways_for_nsteps(int n)
{
if (n <= 1) return 1;
if (n <= 2) return 2; // 1,1; 2
if (n <= 3) return 4; // 1,1,1; 1,2; 2,1; 3

if (result[n]) return result[n];

result[n] = howmany_ways_for_nsteps(n-1)
+ howmany_ways_for_nsteps(n-2)
+ howmany_ways_for_nsteps(n-3);

return result[n];
}
unsigned long long howmany_ways_to_pass_through_100Steps( void )
{
memset(result, 0, sizeof result);

return howmany_ways_for_nsteps(N);
}

int main( int argc, char* argv[] )
{
printf("%llu",
howmany_ways_to_pass_through_100Steps()
);

return 0;
}

// 2229828423