The problem asks for a formula to find the number of squares a diagonal passes through in a rectangle. The example in the image shows a 6 by 4 rectangle, with the diagonal passing through several squares.

This is a classic problem in geometry, and the solution involves a well-known formula:

Formula:

The number of squares that the diagonal of an $m \times n$ rectangle passes through is given by:

$\text{Number of squares} = m + n - \gcd(m, n)$

Where:

• $m$ is the number of rows (or the height of the rectangle),
• $n$ is the number of columns (or the width of the rectangle),
• $\gcd(m, n)$ is the greatest common divisor of $m$ and $n$.

Applying the formula to a 6x4 rectangle:

• $m = 6$
• $n = 4$
• $\gcd(6, 4) = 2$

So, the number of squares the diagonal passes through is:

$6 + 4 - 2 = 8$

Thus, the diagonal of the 6 by 4 rectangle passes through 8 squares.

Would you like further clarification or additional examples on this topic?

Here are 5 related questions to deepen your understanding:

1. How would the formula change if the rectangle were a square, such as 5x5?
2. Can you derive this formula from scratch using concepts of number theory and geometry?
3. What happens when the dimensions $m$ and $n$ are coprime (i.e., $\gcd(m, n) = 1$)?
4. How does the formula behave when either $m$ or $n$ is very large?
5. How can you generalize this formula to work in higher-dimensional spaces, like cubes?

Tip: The greatest common divisor (GCD) plays a key role in this formula because it helps account for the overlap of diagonal lines along grid points.