25
The number of balls to make an equilateral triangle with the sides of length n is expressed by the formula n(n+1)/2
So to express the number of balls you have is
b = n(n+1)/2 + 4
With the larger arrangement you have
b = (n+1)(n+2)/2 - 3
Both of those qualities are equal to each other, so set an equation where they're equal. Then solve for n
n(n+1)/2 + 4 = (n+1)(n+2)/2 - 3
Distribute the n term on the left
(n^2 + n)/2 + 4 = (n+1)(n+2)/2 - 3
Distribute the /2 on the left.
0.5n^2 + 0.5n + 4 = (n+1)(n+2)/2 - 3
Multiply (n+1)(n+2) on right, then distribute the /2
0.5n^2 + 0.5n + 4 = 0.5n^2 + 1.5n + 1 - 3
Subtract 0.5n^2 + 0.5n from both sides
4 = n + 1 - 3
Add 2 to each side
6 = n
So the original triangle had sides with a length of 6, for a total number of balls of
6(6+1)/2 = 21
with 4 extra giving 25 balls. Let's check with the next larger triangle
7(7+1)/2 = 28
with 3 balls shortage. Which 25 balls would make happen.
So the number of balls in the set is 25
