The longest side of a triangle is three times the shortest side. The third side is $2$ cm shorter than the longest side. If the perimeter is at least $61$ cm, what is the **minimum** length of the shortest side?

Let $x$ be the shortest side. Longest $= 3x$. Third $= 3x - 2$. Perimeter: $x + 3x + (3x - 2) = 7x - 2 \geq 61 \Rightarrow 7x \geq 63 \Rightarrow x \geq 9$. Minimum shortest side $= \boxed{9}$ cm. Check at $x = 9$: sides $9, 27, 25$. Perimeter $= 61$ ✓. Triangle inequality: $9 + 25 = 34 > 27$ ✓.