How many Triangles?

Probability Level 2

This pattern has 6 levels of triangles. What is the total number of triangles in this pattern? Similarly, try to find out a general solution for total number of triangles where number of levels is even more.

The answer is 78.

2 solutions

Utkarsh Duvey
Apr 8, 2018

It can be derived that Number of up triangle = n(n+1)(n+2)/6 Number of down triangle = (n-2)(n-3)(4n-13)/6 Putting n=6, we get total = 56+22 = 78

Roy Arijit
Oct 20, 2014

Here's the procedure to find the total number of triangles.. First calculate the number of levels.. For this image, number of level is 6.

Calculating number of triangle pointing upwards:\

Size = 6: No of triangle = 1 Size = (6-1) : No of triangles = (1+2) Size = (6-2) : No of triangles = (1+2+3) . . . Size = (6-5) : No of triangles = (1+2+3+4+5+6)

Calculating number of triangles pointing downwards:

maximum size of triangle pointing downwards = floor(number of level/2) here it's floor(6/2) = 3 Size = 1 : No of triangles = (1+2+3+4+5) Size = 2 : No of triangles = (1+2+3) Size = 3 : No of triangles = 1 (pick the last but one number from the stack of numbers for calculating no of triangles pointing upwards, skip the next number and pick the next number. Continue till reaching the first number)

Here's the pseudo-code

No of level=n,m=0,k=0; For(i=1 to n){ For (j=1 to i){ k=k+j; } a[i]=k; m=m+k; k=0; } //Calculating triangles pointing upwards For (i=n-1 to 1) { m=m+a[i]; i=i-2; } //Adding triangles pointing downwards Total no of triangles=m; //answer