Grid boundary vs interior tiles

CAT 2005 Slot 1 · QA · Medium · Algebra

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A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is

  1. A.

    10

  2. B.

    12

  3. C.

    14

  4. D.

    16

Answer

B

Explanation

Let the floor have m×nm \times n tiles. Interior (red) tiles =(m−2)(n−2)= (m-2)(n-2). Boundary (white) tiles =mn−(m−2)(n−2)= mn - (m-2)(n-2). Given red = white: (m−2)(n−2)=mn−(m−2)(n−2)  ⟹  (m−2)(n−2)=12mn(m-2)(n-2) = mn - (m-2)(n-2) \implies (m-2)(n-2) = \frac{1}{2} mn. 2(mn−2m−2n+4)=mn  ⟹  mn−4m−4n+8=0  ⟹  (m−4)(n−4)=82(mn - 2m - 2n + 4) = mn \implies mn - 4m - 4n + 8 = 0 \implies (m-4)(n-4) = 8. Possible factor pairs for 8: (1,8)(1, 8) or (2,4)(2, 4).

  • (1,8)  ⟹  m=5,n=12(1, 8) \implies m = 5, n = 12.
  • (2,4)  ⟹  m=6,n=8(2, 4) \implies m = 6, n = 8. So possible edge sizes are 5, 6, 8, 12. Among the options, 12 is present.

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