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
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