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How many ways to cover this rectangle

  1. Oct 2, 2012 #1
    1. The problem statement, all variables and given/known data
    In how many different non-overlapping ways can a (2 x 10) rectangle be covered by (1 x 1) and (1 x 3) rectangles.

    2. Relevant equations
    3. The attempt at a solution

    I've never done any question like this. The only solution I can think of is a brute force method but I doubt that's the way it is intended to be solved. Any hints? Thanks.
     
  2. jcsd
  3. Oct 2, 2012 #2

    Mark44

    Staff: Mentor

    Since the smaller rectangles can't overlap the larger (2 x 10) rectangle, you can simplify matters by looking at how you would cover a 1 x 10 rectangle with 1 x 1 squares and 1 x 3 rectangles.
     
  4. Oct 3, 2012 #3
    That's true! Thanks. So if I did it correctly this what I got for a 1 x 10 rectangle. (For simplicity, rectangle refers to 1x3 and square refers to 1x1)


    0 rectangles and 10 squares: 1
    1 rectangle and 7 squares: 8
    2 rectangle and 4 squares: 15
    3 rectangle and 1 squares: 4


    Total = 28 ways to cover a 1 x 10 rectangle.

    How would I extrapolate from this to a 2 x 10 rectangle? Would it be 282? Thanks.
     
    Last edited: Oct 3, 2012
  5. Oct 3, 2012 #4

    Mark44

    Staff: Mentor

    I haven't checked your figures, but for the 2 x 10 rectangle, 272 seems reasonable. For each of the 27 ways on one of the 1 x 10 strips, there are 27 different arrangements on the other strip.
     
  6. Oct 4, 2012 #5
    I initially made a mistake (missed one arrangement). Right now I have 28. Thanks. (Previous post has been edited)
     
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