Cube Dimensions and Original Surface Area

To begin, let's determine the dimensions of the main cube. The question states that the cube is constructed using 64 smaller cubic blocks, each with a side length of one unit.

To find the side length of the larger cube, we calculate the cube root of the total number of small blocks:

Side length of the large cube = $\sqrt[3]{\text{Total cubic blocks}}$

Side length of the large cube = $\sqrt[3]{64}$

Side length of the large cube = $4$ units.

Therefore, the main cube has dimensions of $4 \times 4 \times 4$ units.

Next, we calculate the original surface area of this large cube. The formula for the surface area of a cube is:

Surface Area = $6 \times (\text{Side length})^2$

Original Surface Area = $6 \times (4 \text{ units})^2$

Original Surface Area = $6 \times 16 \text{ square units}$

Original Surface Area = $96 \text{ square units}$.

Corner Removal Impact on Surface Area

The problem specifies that one cubic block is removed from every corner of the cube. A standard cube has 8 corners. Let's analyze how the removal of a single 1x1x1 corner block affects the overall surface area.

• Initially, each corner block exposes three of its faces to the outside, each with an area of $1 \times 1 = 1$ square unit. So, removing this block means these three outer faces are no longer part of the surface. This would seemingly lead to a reduction of $3 \times 1 = 3$ square units in surface area.
• However, when this 1x1x1 corner block is taken out, it reveals three new faces on the interior of the main cube that were previously hidden. These newly exposed faces also have an area of $1 \times 1 = 1$ square unit each. Thus, these new internal faces add $3 \times 1 = 3$ square units to the total surface area.

Considering both the loss and gain, the net change in surface area for each corner where a block is removed is:

Net change per corner = (Area lost from original exposed faces) + (Area gained from new exposed faces)

Net change per corner = $-3 \text{ square units} + 3 \text{ square units} = 0 \text{ square units}$.

Total Surface Area After Block Removal

Since there are 8 corners in the cube, and the removal of a block from each corner results in a net change of 0 square units to the surface area, the total surface area of the body remains unaffected after all corner blocks are removed.

Total number of corners = 8

Total change in surface area = $8 \times (\text{Net change per corner})$

Total change in surface area = $8 \times 0 \text{ square units} = 0 \text{ square units}$.

Therefore, the resulting surface area of the body is exactly the same as its original surface area.

Resulting Surface Area = Original Surface Area = $96 \text{ square units}$.