Pattern Recognition - 33

There is a competition held for school students where each team of 20 students are given 66 balls. They need to arrange the marbles in a triangle shape. Something like this.

Each student in a team can fill only one row. They have to decide the number of students needed to complete the shape with the given no of marbles(which is 66), beforehand. Can you help them find out how many students (out of 20) need to take part in the game to win it?  


Answer

The idea here is quite simple. We need to arrange the balls in triangle shape. Have you heard about triangular number series? 

Tn  (n th triangular number) =  [ n* (n+1)] /2

1st triangular number - (1* 2)/2   = 1
2nd triangular number - (2*3)/2   = 3
3rd triangular number - (3*4)/2   = 6
4th triangular number - (4* 5)/2  = 10
5th triangular number - (5* 6)/2  = 15

1 , 3 , 6 , 10 , 15 ,......................................................

So, the value of n in Tn will be numbers of rows in the triangular shape.
i.e : 1st triangular number will have 1 row , 2nd will have 2 rows, 3rd will have 3 rows....likewise....

To arrange the balls in a triangle shape we need to find n where Tn=66

I think the easiest way is to  make a rough guess and find it  out.

or  find all the triangular numbers till you come to 66.
In that way 1 , 3 , 5 , 10 , 15 , 21 , 28 , 36 , 45, 55  , 66 ,.......................

66 comes as the 11th number in the series.

11th triangular number means there are 11 rows so we need 11 students.