Let's assume the original side length of the cube is "x" cm.
The volume of the original cube is given by V = x^3 cubic cm.
If a 3 cm slice is cut off from one side, the new side length will be (x - 3) cm. The volume of the remaining portion is given as 17,496 cubic cm.
So, the new volume can be expressed as (x - 3)^3 = 17,496.
Taking the cube root of both sides, we get x - 3 = 26.
Adding 3 to both sides, we get x = 29.
Therefore, the original cube had side length 29 cm.