Counting Rectangles Cut By a Diagonal

Date: 06/15/99 at 13:27:47
From: Ben
Subject: Rectangles Cut By a Diagonal Line

I am writing to ask you if you can help me on this project we have 
been given. What we are asked to do is find an equation for the number 
of 1 cm squares that are cut by a line going through the middle (from 
corner to corner.)

For example, with a 3 x 5 rectangle (each square = 1 cm) the number of 
rectangles cut is 7.

Please can you try to help us? Thanks!

Date: 06/15/99 at 17:38:05
From: Doctor Anthony
Subject: Re: Rectangles Cut By a Diagonal Line

The general rule: If the lengths of sides (a x b) of the rectangle are 
mutually prime, the number of squares cut is a+b-1

Thus, your example: (3 x 5) gives 3+5-1 = 7

Other examples:     (8 x 5) gives 8+5-1 = 12
                    (9 x 4) gives 9+4-1 = 12
                    (9 x 5) gives 9+5-1 = 13

BUT (9 x 6) DOES NOT give 9+6-1 = 14. Instead you must proceed as 
follows:

   First divide (9 x 6) through by common factor 3 to get (3 x 2)
   Then apply the rule to (3 x 2) to give 3+2-1 = 4
   Now multiply by the factor 3 again to get 12 (which is correct).

Let's do a square figure, say (5 x 5). We divide through by 5. This 
gives (1 x 1). Applying the rule gives 1+1-1 = 1. Now multiply up by 
the factor 5 again to get 5. We know this is correct because in ANY 
square figure the number of squares that are cut will be equal to the 
side of the square.

The general procedure for a rectangle (a x b) is as follows:

   If a and b are relatively prime the answer is a+b-1

   If a and b have a common factor c, first divide through by c to get
     (a/c x b/c).
   Then apply the rule to get  a/c + b/c - 1.
   Finally, multiply through again by c to get (a+b-c).

For example, with (9 x 6) the correct answer is 9+6-3 = 12.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/