Some identical balls are arranged in rows to form an equilateral triangle: the first row h…

Question

Some identical balls are arranged in rows to form an equilateral triangle: the first row has $1$ ball, the second $2$, and so on. If $99$ more balls are added to the total, they can be arranged in a square whose each side has exactly $2$ balls less than each side of the triangle. Then the number of balls used to form the triangle is:

Accepted Answer

Triangle with $n$ balls per side has $\dfrac{n(n+1)}{2}$ balls; square side is $n-2$.

$$\frac{n(n+1)}{2}+99 = (n-2)^2 \implies n^2-9n-190=0 \implies (n-19)(n+10)=0 \implies n=19.$$

$$\text{Balls in triangle} = \frac{19\times20}{2} = 190.$$

Solutions · 1