Problem: A square of side 4cm and uniform thickness is divided into four equal squares. If one of the square is cut off, find the center of mass of the remaining portion.

Solution:
To find the center of mass of the remaining portion, we need to first find the center of mass of the whole square and then subtract the center of mass of the cut-off square.

Finding the Center of Mass of the Whole Square:
Since the square is uniform, the center of mass will be at its geometrical center. The geometrical center of the square is given by the intersection of its diagonals. The diagonals of the square are of equal length and intersect at a right angle.

Therefore, the center of mass of the whole square will be at the point where the diagonals intersect, which is also the midpoint of each diagonal. Let's call this point O.

Finding the Center of Mass of the Cut-Off Square:
The cut-off square has a side length of 2cm. To find its center of mass, we need to find the midpoint of its diagonal. Let's call this point A.

The diagonal of the cut-off square is equal to the side length of the original square, which is 4cm. Therefore, we can use the Pythagorean theorem to find the length of the diagonal of the cut-off square, which is √(2^2 + 2^2) = 2√2 cm.

Since the diagonal of the cut-off square passes through its center, A, it divides the cut-off square into two congruent right triangles. The midpoint of the diagonal, A, is also the centroid of each triangle. Therefore, the center of mass of the cut-off square is at point A.

Finding the Center of Mass of the Remaining Portion:
To find the center of mass of the remaining portion, we need to subtract the center of mass of the cut-off square from the center of mass of the whole square.

Let's call the remaining portion of the square B. Since the remaining portion consists of three squares, each with a side length of 2cm, the center of mass of B will be the centroid of an equilateral triangle with side lengths of 2cm.

The centroid of an equilateral triangle is located at a distance of 1/3 of the height from the base. Therefore, the center of mass of B will be located at a height of 2√3/3 cm from the bottom of the remaining square.

To find the horizontal position of the center of mass of B, we need to find the distance between the midpoint of the diagonal of the cut-off square and the center of mass of the whole square. This distance is equal to half the side length of the whole square, which is 2cm.

Therefore, the horizontal position of the center of mass of B will be at a distance of 2cm to the right of the midpoint of the diagonal of the cut-off square.

Final Answer:
The center of mass of the remaining portion is located at a height of 2√3/3 cm from the bottom of the remaining square and at a distance of 2cm to the right of the midpoint of the diagonal of the cut-off square.