S2T2

Jim Randell 10:55 am on 3 January 2023 Permalink Reply
Tags: by: C Higgins ( 35 ), by: H Bradley ( 35 )   

• 

  Jim Randell 10:56 am on 3 January 2023 Permalink | Reply

  This Python program generates candidate areas for the red, green and blue rectangles. It then looks for appropriate dimensions that satisfy the remaining conditions, and allow them to fit together as specified.

  It runs in 60ms. (Internal runtime is 7.3ms).

  Run: [ @replit ]

  from enigma import (irange, nrev, div, divisors_pairs, cproduct, ordered, printf)
  
  # fit the rectangles (x1, y1), (x2, y2) and a rectangle area A together
  def fit(x1, y1, x2, y2, A):
    for z1 in {x1, y1}:
      # can we match a side?
      if z1 in {x2, y2}:
        (x, y) = (z1, div(A, z1))
        if y is not None:
          yield ordered(x, y)
      # can we join sides?
      for z2 in {x2, y2}:
        y = (x1 + y1 - z1) - (x2 + y2 - z2)
        if y > 0:
          if z2 * y == A:
            yield ordered(z2, y)
        elif y < 0:
          y = -y
          if z1 * y == A:
            yield ordered(z1, y)
  
  # consider the area of the flag, it is a 3-digit number
  for A in irange(100, 999):
    # and the area of blue is the reverse of this
    B = nrev(A)
    if B < 100 or not (B < A): continue
    # the area of red and green are 1/3 and 2/3 of what is left
    R = div(A - B, 3)
    if R is None: continue
    G = 2 * R
  
    # find dimensions for red and green, such that red has twice the perimeter of green
    for ((rx, ry), (gx, gy)) in cproduct(divisors_pairs(v) for v in (R, G)):
      if not (rx + ry == 2 * (gx + gy)): continue
  
      # find dimensions of blue that fits to make a rectangle
      for (bx, by) in fit(rx, ry, gx, gy, B):
        # output solution
        printf("A={A} [R={R} G={G} B={B}]", R=(rx, ry), G=(gx, gy), B=(bx, by))
  

  Solution: The area of the flag is: 231 sq cm.

  The flag is a 7×33 rectangle, as shown below:

  

  The red rectangle is: 1×33. The green rectangle is 6×11. The blue rectangle is 6×22.

  (Dimensions in cm).

  ---

  We have sometime come across claims that a phrase such as “a whole number of pounds” (or similar – see Teaser 2869, Teaser 2692) necessarily implies a value must be at least 2 (although I do not accept this argument myself). But this puzzle uses “a whole number of centimetres” to refer to a value of 1, and without allowing this value there would be no solution.

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• 

  GeoffR 3:28 pm on 3 January 2023 Permalink | Reply

  % A Solution in MiniZinc - same rectangle arrangement
  % Red = A * B, Green = C * D, Blue = C  * E, where B = D + E
  include "globals.mzn";
  
  % Assumed Rectangle/Area dimension ranges
  var 1..25:A; var 1..50:B;
  var 1..25:C; var 1..25:D; var 1..25:E;
  
  var 10..999:flag_area;
  var 1..999:red_area; var 1..999:green_area; var 1..999:blue_area;
  var 4..200:red_perim; var 4..200:green_perim;
  
  % Red length = Green length + Blue length
  constraint B == D + E;
  
  % Digits (x,y,z) for 3-digit flag area
  var 1..9:x; var 0..9:y;  var 1..9:z; 
  
  % Find three digits of flag area
  constraint x == flag_area div 100 /\ y == flag_area div 10 mod 10
  /\ z == flag_area mod 10;
  
  % Flag area digits are reversed for the blue area
  constraint blue_area = C * E /\ blue_area == 100*z + 10*y + x; 
  constraint green_area = C * D /\ red_area = A * B;
  constraint red_perim = 2 * (A + B);
  constraint green_perim == 2 * (C + D);
  
  % Area/perimeter relationships
  constraint green_area = 2 * red_area;
  constraint red_perim = 2 * green_perim;
  constraint flag_area = red_area + blue_area + green_area;
  
  solve satisfy;
  
  output [ "Flag area = " ++ show(flag_area) ++ " sq. cm.," 
  ++ " Red area = " ++ show(red_area) ++ " sq. cm." 
  ++ "\n" ++ "Green area = " ++ show(green_area) ++ " sq. cm.,"
  ++ " Blue area = " ++ show(blue_area) ++ " sq. cm."  ];
  
  % Flag area = 231 sq. cm., Red area = 33 sq. cm.
  % Green area = 66 sq. cm., Blue area = 132 sq. cm.
  % ----------
  % ==========
  
  
  

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