Good question.

As expected, this question boils down to 2 equation,

Consider total number of Balls in Triangle = T and number of balls in last row = x.

1+2+3+...+x = T

x(x+1)/2 = T ----(a)

As mentioned in the question, side of a square will be (x-8) and total number of Balls in square will be (T+669)

(x-8)^2 = T+669 -----(b)

Now the hardest part of the question will be to solve these 2 equations and this looks like time consuming BUT the easy way will be plug and play. Also, we've to find a value of T (from 5 optiosn given below) which can make a square of a a number. One we know this, it will be a cake walk.

We can see that option A fits this criteria in eq (b). Add - 1540+669 = 2209 = 47^2 = (x-8)^2 Hence, x = 55.

Cross check by putting in eq (a) = x(x+1)/2 = T = > 56*55=1540*2.

Hence, answer is A.

Hope it helps.

Cheers!

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