Hello there, I'm Kimi, a specialist in mathematical puzzles and logical reasoning. I'm here to help you unravel the mysteries of the 8x8 checkerboard and its rectangles.

Let's dive into the problem at hand. We're looking to find out how many rectangles can be formed on an 8x8 checkerboard. A rectangle, as defined, is a shape with four right angles and opposite sides that are equal in length. On a checkerboard, these sides are formed by the intersections of the horizontal and vertical lines.

Now, the initial thought might be to consider the number of ways to choose two vertical lines and two horizontal lines, as suggested. However, this approach overlooks the fact that the length and width of the rectangles can vary, and not all combinations of lines will form a rectangle.

To accurately count the rectangles, we need to consider the possible lengths of the rectangles' sides. The smallest rectangle you can form on an 8x8 checkerboard has sides of length 1 (a 1x1 square), and the largest has sides of length 8 (an 8x8 square). For each possible length of one side, there are several ways to choose the other side's length.

Let's break it down:

- For a 1x1 square, there are 8x8 = 64 possible squares.
- For a 1x2 rectangle, there are 7 ways to choose the vertical side and 7 ways to choose the horizontal side, resulting in 7x7 = 49 rectangles.
- This pattern continues, with the number of ways to choose the sides decreasing as the length of the sides increases.

To generalize, for a rectangle with a width of \( w \) and a length of \( l \), where \( 1 \leq w, l \leq 8 \), the number of rectangles is \( (8-w+1) \times (8-l+1) \). This is because there are \( (8-w+1) \) ways to place the width and \( (8-l+1) \) ways to place the length.

Adding these up for all possible widths and lengths gives us the total number of rectangles:

\[
\sum_{w=1}^{7} \sum_{l=w+1}^{8} (8-w+1) \times (8-l+1)
\]

Calculating this sum, we get:

\[
(7 \times 1 + 6 \times 2 + 5 \times 3 + 4 \times 4 + 3 \times 5 + 2 \times 6 + 1 \times 7) \times 8 + 64
\]

\[
= (7 + 12 + 15 + 16 + 15 + 12 + 7) \times 8 + 64
\]

\[
= 94 \times 8 + 64
\]

\[
= 752 + 64
\]

\[
= 816
\]

So, there are 816 rectangles on an 8x8 checkerboard.

Now, let's move on to the translation.

read more >>

A rectangle is formed by two vertical and two horizontal lines. On an 8x8 chessboard, you can choose two vertical lines in C(9, 2) = 9!/(2!*7!) = 36 different ways. Similarly you can choose two horizontal lines in 36 different ways. So, the answer is 36*36 = 1296.read more >>