S2T2

Jim Randell 3:23 pm on 18 August 2019 Permalink Reply
Tags: by: Len Wood   

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  Jim Randell 3:24 pm on 18 August 2019 Permalink | Reply

  Instead of considering cutting up the rectangle we consider adding squares to an existing rectangle.

  Tina’s 4th and 5th squares are 1 inch squares, so come from a 2 × 1 rectangle. A square can be added to any rectangle by extending it horizontally or vertically, and we need to add three more squares, so there are 8 possible starting rectangles.

  Cindy’s rectangle is also one of these 8 appropriately scaled to make the rectangles have the same area. If Cindy’s rectangle is scaled by f along each side then the area is scaled by f².

  Also the area of Tina’s first square is twice the area of Cindy’s first square.

  This Python 3 program finds the 8 possible rectangles and their divisions and finds a pair that give an appropriate scale factor.

  It runs in 87ms.

  Run: [ @repl.it ]

  from enigma import subsets, printf
  
  # extend the rectangle (x, y) with k more squares
  def solve(x, y, k, rs=[]):
    if k == 0:
      yield (x, y, rs)
    else:
      # grow in the x direction by adding a y square
      yield from solve(x + y, y, k - 1, rs + [y])
      # grow in the y direction by adding an x square
      yield from solve(x, x + y, k - 1, rs + [x])
  
  # choose rectangles for T and C
  for ((Tx, Ty, Tss), (Cx, Cy, Css)) in subsets(solve(1, 2, 3, [1, 1]), size=2, select="P"):
    (A, f2A) = (Tx * Ty, Cx * Cy)
    # check scale factors
    if f2A * Tss[-1] ** 2 == 2 * A * Css[-1] ** 2:
      # output solution
      printf("area = {A} [T: {Tx}x{Ty} = {Tss}, fC: {Cx}x{Cy} = {Css}]", A=Tx * Ty)
  

  Solution: The area of the rectangles was 28 square inches.

  Tina’s rectangle was 4×7, and her squares had area: 16, 9, 1, 1, 1.

  Cindy’s rectangle was (7√2)×(2√2), and her squares had area: 8, 8, 8, 2, 2.

  Here is a diagram of the two rectangles:

  

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