Geometry

A standard chessboard is an 8x8 grid, but we need to find the total number of squares of all sizes that can be formed on the chessboard.
Step 1: Count of 1x1 squares.
The number of 1x1 squares is simply the number of small squares on the board: 8 × 8 = 64 1x1 squares.
Step 2: Count of 2x2 squares.
To form a 2x2 square, we can choose the top-left corner from any of the first 7 rows and 7 columns (since a 2x2 square takes up two rows and two columns). Thus, there are: 7 × 7 = 49 2x2 squares.
Step 3: Count of 3x3 squares.
Similarly, for 3x3 squares, we can choose the top-left corner from any of the first 6 rows and 6 columns. Thus, there are: 6 × 6 = 36 3x3 squares.
Step 4: Count of 4x4 squares.
For 4x4 squares, we can choose the top-left corner from any of the first 5 rows and 5 columns. Thus, there are: 5 × 5 = 25 4x4 squares.
Step 5: Count of 5x5 squares.
For 5x5 squares, we can choose the top-left corner from any of the first 4 rows and 4 columns. Thus, there are: 4 × 4 = 16 5x5 squares.
Step 6: Count of 6x6 squares.
For 6x6 squares, we can choose the top-left corner from any of the first 3 rows and 3 columns. Thus, there are: 3 × 3 = 9 6x6 squares.
Step 7: Count of 7x7 squares.
For 7x7 squares, we can choose the top-left corner from any of the first 2 rows and 2 columns. Thus, there are: 2 × 2 = 4 7x7 squares.
Step 8: Count of 8x8 squares.
Finally, there is only 1 square of size 8x8, which is the entire chessboard itself: 1 × 1 = 1 8x8 square.
Step 9: Total number of squares.
The total number of squares is the sum of all the squares of different sizes: 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 total squares.
Therefore, the correct answer is (3) 204.

Step 1: Understanding the geometry of the problem.
The equilateral triangle has a side length of 6 cm. When we cut off the corners, we form a regular hexagon inside the triangle. The regular hexagon is inscribed in the triangle, and its sides are determined by cutting off the corners of the equilateral triangle.

Step 2: Area of regular hexagon.
To find the area of the regular hexagon, we use the formula:

Ahexagon = (3√3/2) s2

where s is the side length of the hexagon.
Since the hexagon is inscribed inside the equilateral triangle, the side of the hexagon corresponds to the side of the smaller triangles that are cut off from the corners.

Step 3: Side length of the hexagon.
For a regular hexagon inscribed inside an equilateral triangle, the side length of the hexagon is the same as the side length of the smaller triangles formed by cutting the corners. This side length is equal to s/2, where s is the side length of the equilateral triangle.
So, the side length of the hexagon is:

shexagon = 6 cm

Step 4: Calculate the area of the hexagon.
Now, we calculate the area of the regular hexagon:

Ahexagon = (3√3/2) (6)2 = (3√3/2) × 36 = 6√3 cm2

Step 5: Conclusion.
The area of the regular hexagon formed by cutting off the corners of the equilateral triangle is 6√3 cm².
Thus, the correct answer is (3) 6√3 cm².