A puzzle related to nested loops - loops

Question

For a given input N, how many times does the enclosed statement executes?
for i in 1 … N loop
for j in 1 … i loop
for k in 1 … j loop
sum = sum + i ;
end loop;
end loop;
end loop;
Can anyone figure out an easy way or a formula to do this in general. Please explain.

Answer 1

First, I written a C code to generate sum:
int main(){
int i =0, k =0, j =0, n =0;
int N =0;
int sum =0;
N =10;
for (n=1; n <= N; n++){
// unindented code here
sum =0;
for (i=1; i<=n; i++)
for (j=1; j<=i; j++)
for (k=1; k<=j; k++)
sum++;
printf("\n N=%d sum = %d",n, sum);
}
printf("\n");
}
Simple compile and generate result for N=1 to N=10 :
$ gcc sum.c
$ ./a.out
N=1 sum = 1
N=2 sum = 4
N=3 sum = 10
N=4 sum = 20
N=5 sum = 35
N=6 sum = 56
N=7 sum = 84
N=8 sum = 120
N=9 sum = 165
N=10 sum = 220
Then, Tried to explore How this works? with some diagrams:
For, N=1:
i<=N, (i=1)
|
j<=i, (j=1)
|
k<=j, (K=1)
|
sum=0. sum++ ---> sum = 1
That is (1) = 1
For, N=2:
i<=N, (i=1)-------(i=2)
| |-----|-----|
j<=i, (j=1) (j=1) (j=2)
| | |----|----|
k<=j, (K=1) (K=1) (K=1) (K=2)
| | | |
sum=0, sum++ sum++ sum++ sum++ --> sum = 4
That is (1) + (1 + 2) = 4
For, N=3:
i<=N, (i=1)-------(i=2)--------------------(i=3)
| |-----|-----| |---------|-------------|
j<=i, (j=1) (j=1) (j=2) (j=1) (j=2) (j=3)
| | |----|----| | |----|----| |-----|-----|
k<=j, (K=1) (K=1) (K=1) (K=2) (K=1) (K=1) (K=2) (K=1) (K=2) (K=3)
| | | | | | | | | |
sum=0, sum++ sum++ sum++ sum++ sum++ sum++ sum++ sum++ sum++ sum++
--> sum = 10
That is (1) + (1 + 2) + ( 1 + 2 + 3 ) = 10
N = 1, (1) = 1
N = 2, (1) + (1 + 2) = 4
N = 3, (1) + (1 + 2) + (1 + 2 + 3) = 10
N = 4, (1) + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) = 20
N = 5, (1) + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + (1 + 2 + 3 + 4 + 5) = 35
Finally, I could understood that sum of N in three loop is:
(1) + (sum 0f 1 to 2) + ... + (sum of 1 to (N-2)) + (sum of 1 to (N-1) ) + (sum of 1 to N)
or we can write it as:
=> (1) + (1 + 2) + ...+ (1 + 2 +....+ i) + ... + (1 + 2 + ....+ N-1) + (1 + 2 + ....+ N)
=> ( N * 1 ) + ( (N-1) * 2) + ( (N-2) * 3) +...+ ( (N -i+1) * i ) +... + ( 1 * N)
You can refer here for simplification calculations: (I asked HERE )
[YOUR ANSWER]
= ( ((N) * (N+1) * (N+2)) / 6 )
And, I think its correct. I checked as follows:
N = 1, (1 * 2 * 3)/6 = 1
N = 2, (2 * 3 * 4)/6 = 4
N = 3, (3 * 4 * 5)/6 = 6
N = 4, (4 * 5 * 6)/6 = 10
N = 5, (5 * 6 * 7)/6 = 35
Also, The complexity of this algorithm is O(n3)
EDIT:
The following loop also has same numbers of count, that is = ( ((N) * (N+1) * (N+2)) / 6 )
for i in 1 … N loop
for j in i … N loop
for k in j … N loop
sum = sum + i ;
end loop;
end loop;
end loop;