To determine how many squares can be formed on a chessboard with 6 horizontal and 6 vertical lines, let's break it down.

Explanation:

• The chessboard has 6 horizontal lines and 6 vertical lines, which means it forms a grid.
• The size of a square is determined by selecting two horizontal lines and two vertical lines, because any pair of horizontal and vertical lines can form the boundary of a square.

Step-by-Step Calculation:

1. 1x1 squares: Each small square is formed by two adjacent horizontal and two adjacent vertical lines.

   • There are $5 \times 5 = 25$ such squares (as the space between lines forms 5 small squares along each dimension).

2. 2x2 squares: To form a 2x2 square, we need two horizontal lines that are 2 units apart and two vertical lines that are 2 units apart.

   • There are $4 \times 4 = 16$ such squares.

3. 3x3 squares: Similarly, a 3x3 square is formed by selecting horizontal and vertical lines 3 units apart.

   • There are $3 \times 3 = 9$ such squares.

4. 4x4 squares: For a 4x4 square, the lines must be 4 units apart.

   • There are $2 \times 2 = 4$ such squares.

5. 5x5 square: Finally, one square is formed by selecting the outermost horizontal and vertical lines.

   • There is $1 \times 1 = 1$ such square.

Total number of squares:

Now, sum all the squares: $25 + 16 + 9 + 4 + 1 = 55$

Thus, 55 squares can be formed on this 5x5 grid.

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Related Questions:

1. How would the number of squares change if there were 7 horizontal and 7 vertical lines?
2. How many rectangles can be formed on a chessboard with 6 horizontal and 6 vertical lines?
3. How can the formula for counting squares on a grid be generalized for any $n \times n$ grid?
4. What is the number of squares on a standard 8x8 chessboard?
5. How many rectangles (including squares) can be formed on a standard chessboard?

Tip:

For any $n \times n$ grid, the number of squares is the sum of the squares of the integers from 1 to $n-1$.