Diagonals and Tiles

Date: 11/17/2001 at 22:07:53
From: Kamie Oda
Subject: Diagonal and Relatively Prime

Dear Dr. Math,

Jay tiled a 15 feet by 21 feet rectangular ballroom with one-foot-
square tiles. When he finished, he drew both diagonals 
connecting opposite corners of the room. What is the total number 
of tiles that the diagonals passed through?

I used a procedure from a similar problem from the archives, but 
got the wrong answer. Can you help me find my mistake? 15x21, 
greatest common divisor is 3, 5+7-1 = 11, 11x3 = 33, 33x 2 diagonals 
= 66 tiles. According to my book, the answer is 63. How can the 
answer be an odd number when there are 2 diagonals?

Thanks for the help!  
Kamie

Date: 11/18/2001 at 12:29:20
From: Doctor Sarah
Subject: Re: Diagonal and Relatively Prime

Hi Kamie - thanks for writing to Dr. Math.

You need to be careful not to double-count tiles. Let's look at a 
diagram:

   

Tiles with diagonals passing through them are shaded.  Do you 
see any tiles with more than one diagonal?  Those tiles should 
only be counted once.

For more information from the Dr. Math archives, see:

   Using Relative Primes
   http://mathforum.org/dr.math/problems/chin12.7.96.html   

   Basketball Court
   http://mathforum.org/dr.math/problems/sadeghi1.7.98.html   

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

Date: 11/18/2001 at 15:06:18
From: Marlene Oda
Subject: Re: Diagonal and Relatively Prime

Dear Doctor Sarah,

Thank you for clearly showing me where my extra tiles came 
from. I had tried drawing diagonals on graph paper, but my drawing 
wasn't accurate and I failed to see the overlap. Thank you for 
taking the time to provide the illustration, which clearly showed 
3 overcounted tiles! Therefore the answer is 63 not 66. How 
enlightening!

Kamie