The sides of a triangle are equal to integral numbers of units. Two sides are 4 and 6 units long, respectively; what is the minimum value for the triangle's perimeter?

The sides of a triangle must all be greater than zero. The sum of the lengths of the two shorter sides must be greater than the length of the third side. Since we are looking for the minimum value of the perimeter, assume the longer of the two given sides, which is 6, is the longest side of the triangle. Then the third side must be greater than 6 - 4 = 2. Since we are told the sides are all integral numbers, the last side must be 3 units in length. Thus, the minimum length for the perimeter is 4+6+3 = 13 units.