Quandaries and Queries
Who is asking: Student
Question: Draw a rectangle with sides of 3 and 4. Divide the sides into 3 and 4 equal parts respectively. Draw squares joining the points on the sides of the rectangle. You will have 12 small squares inside the 3 x 4 rectangle.
If you draw a diagonal of the rectangle, it will intersect 6 of the the 12 smaller squares.
Similarly, if you have a 4 x 10 rectangle, the diagonal would intersect 12 of the 40 squares inside the rectangle.
Is there an algebric equation that determines the number of squares that will be intersected by the diagonal of a rectangle?
(Number of squares intersected) = 1 + (number of horizontal lines crossed) + (number of vertical lines crossed) - (number of vertices passed through)
Thus for a 3 by 4, there are 2 horizontal, 3 vertical, and no vertices:
For the 4 by 10 there are 3 horizontal, 9 vertical, and one vertex: