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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## A chessboard is an 8X8 array of identically sized squares. ##### This topic has 2 expert replies and 0 member replies ### Top Member ## A chessboard is an 8X8 array of identically sized squares. ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult A chessboard is an 8X8 array of identically sized squares. Each square has a particular designation, depending on its row and column. An L-shaped card, exactly the size of four squares on the chessboard, is laid on the chessboard as shown, covering exactly four squares. This L-shaped card can be moved around, rotated, and even picked up and turned over to give the mirror-image of an L. In how many different ways can this L-shaped card cover exactly four squares on the chessboard? (A) 256 (B) 336 (C) 424 (D) 512 (E) 672 OA B Source: Magoosh ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1440 messages Followed by: 32 members Upvotes: 59 Top Reply BTGmoderatorDC wrote: A chessboard is an 8X8 array of identically sized squares. Each square has a particular designation, depending on its row and column. An L-shaped card, exactly the size of four squares on the chessboard, is laid on the chessboard as shown, covering exactly four squares. This L-shaped card can be moved around, rotated, and even picked up and turned over to give the mirror-image of an L. In how many different ways can this L-shaped card cover exactly four squares on the chessboard? (A) 256 (B) 336 (C) 424 (D) 512 (E) 672 Source: Magoosh $$?\,\,\,:\,\,\,\# \,\,L - {\rm{shaped}}\,\,{\rm{positions}}$$ I was asked if there is a simple, error-avoiding, quick and "natural" approach (just that, LoL) to this very-nice problem. There is! Although we will need to separate the problem in 8 configurations... all of them are trivial: (The time I took to type the solution - drawing included - was approximately 15min. But only 3min to find the solution to myself - ugly-hand-drawing included.) Configuration 1: the "head" (guide-point in red) is up, the "tail" to the right. We have 6 positions for the head in the first left viable column (I have shown the first and last in it) We have 7 positions for the head in each row (I have shown the lower left and the lower right). Multiplicative Principle: 6*7 = 42 possibilities (Numbers 6 and 7 are explained in the first drawing. The others below are analogous.) Configuration 2: the "head" (guide-point in red) is up, the "tail" to the left. We have 6 positions for the head in the first left viable column (I have shown the first and last in it) We have 7 positions for the head in each row (I have shown the lower left and the lower right). Multiplicative Principle: 6*7 = 42 possibilities Configuration 3: the "head" (guide-point in red) is down, the "tail" to the right. We have 6 positions for the head in the first left viable column (I have shown the first and last in it) We have 7 positions for the head in each row (I have shown the lower left and the lower right). Multiplicative Principle: 6*7 = 42 possibilities Configuration 4: the "head" (guide-point in red) is down, the "tail" to the left. We have 6 positions for the head in the first left viable column (I have shown the first and last in it) We have 7 positions for the head in each row (I have shown the lower left and the lower right). Multiplicative Principle: 6*7 = 42 possibilities Configuration 5: the "head" (guide-point in red) is left, the "tail" to the right-down. We have 7 positions for the head in the first left viable column (I have shown the first and last in it) We have 6 positions for the head in each row (I have shown the lower left and the lower right). Multiplicative Principle: 6*7 = 42 possibilities Configuration 6: the "head" (guide-point in red) is right, the "tail" to the left-down. We have 7 positions for the head in the first left viable column (I have shown the first and last in it) We have 6 positions for the head in each row (I have shown the lower left and the lower right). Multiplicative Principle: 6*7 = 42 possibilities Configuration 7: the "head" (guide-point in red) is left, the "tail" to the right-up. We have 7 positions for the head in the first left viable column (I have shown the first and last in it) We have 6 positions for the head in each row (I have shown the lower left and the lower right). Multiplicative Principle: 6*7 = 42 possibilities Configuration 8: the "head" (guide-point in red) is right, the "tail" to the left-up. We have 7 positions for the head in the first left viable column (I have shown the first and last in it) We have 6 positions for the head in each row (I have shown the lower left and the lower right). Multiplicative Principle: 6*7 = 42 possibilities All cases above are exhaustive (i.e, cover all scenarios) and mutually exclusive (i.e., no double-countings), hence: $$? = 8*42 = 336$$ This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br ### GMAT/MBA Expert GMAT Instructor Joined 25 May 2010 Posted: 15203 messages Followed by: 1861 members Upvotes: 13060 GMAT Score: 790 Top Reply BTGmoderatorDC wrote: A chessboard is an 8X8 array of identically sized squares. Each square has a particular designation, depending on its row and column. An L-shaped card, exactly the size of four squares on the chessboard, is laid on the chessboard as shown, covering exactly four squares. This L-shaped card can be moved around, rotated, and even picked up and turned over to give the mirror-image of an L. In how many different ways can this L-shaped card cover exactly four squares on the chessboard? (A) 256 (B) 336 (C) 424 (D) 512 (E) 672 Case 1: Every 6-block rectangle with a base of 2 and a height of 3 yields 4 possible L-shapes: Chessboard: The base for the 6-block rectangle shown above can appear in rows 1 through 6, yielding 6 row options. The bottom left corner for the 6-block rectangle shown above can appear in columns A through G, yielding 7 column options. To combine the 6 row options with the 7 column options, we multiply: 6*7 = 42. Since each of these 42 6-block rectangles will yield 4 possible L-shapes, we get: Total options = 4*42 = 168. Case 2: Every 6-block rectangle with a base of 3 and a height of 2 yields 4 possible L-shapes: Since the chessboard is square, Case 2 must yield the same number of options as Case 1: Total options = 168. Resulting total: Case 1 + Case 2 = 168 + 168 = 336. The correct answer is B. _________________ Mitch Hunt Private Tutor for the GMAT and GRE GMATGuruNY@gmail.com If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon. Available for tutoring in NYC and long-distance. For more information, please email me at GMATGuruNY@gmail.com. Student Review #1 Student Review #2 Student Review #3 Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now. • 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0

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