S2T2

Jim Randell 5:45 pm on 20 August 2021 Permalink Reply
Tags: by: Howard Williams ( 43 )   

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  Jim Randell 6:19 pm on 20 August 2021 Permalink | Reply

  For a throw of dimensions (x, y) there are xy squares, each of which takes 6 units of time to knit.

  Each square has 4 edges, so there are 4xy edges, but the edges on the perimeter are not sewn together, so there are a total of 4xy − 2(x + y) edges that are sewn together, in pairs. And each pair take 5 units of time to sew together.

  Giving a total time for the throw of:

  time(x, y) = 6xy + 5(4xy − 2(x + y))/2
  time(x, y) = 6xy + 5(2xy − x − y)

  This Python program runs in 53ms.

  Run: [ @replit ]

  from enigma import irange, divisors_pairs, printf
  
  # total time for a throw measuring (x, y)
  time = lambda x, y: 6 * x * y + 5 * (2 * x * y - x - y)
  
  # consider the dimensions (x, y) of a "standard" throw (area < 100)
  for a in irange(1, 99):
    for (x, y) in divisors_pairs(a):
      if x == y: continue
      # total time involved
      t = time(x, y)
  
      # consider the time for a square (y, y) throw
      t2 = time(y, y)
      # the average time per square should be 2% longer
      # i.e. t2 / y^2 = 1.02(t / xy) => 100 x t2 = 102 y t
      if 100 * x * t2 == 102 * y * t:
        printf("a={a}; x={x} y={y} t={t}; t2={t2}")
  

  Solution: The standard throw is 5 ft × 7 ft.

  It has an area of 35 sq ft, and takes 500 time units to construct. i.e. 100/7 time units per square.

  A 7 ft × 7 ft throw has an area 49 sq ft, and takes 714 time units to construct. i.e. 102/7 time units per square.

  And we can easily see that this average is 1.02 times the average for a standard throw.

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    Jim Randell 6:39 pm on 20 August 2021 Permalink | Reply

    Solving the equation for the average time per square gives us:

    255y − 245x − 16xy = 0
    y = 245x / (255 − 16x)

    from enigma import irange, div, printf
    
    # calculate the dimension of a standard (x, y) throw
    for x in irange(1, 7):
      y = div(245 * x, 255 - 16 * x)
      if y is not None:
        printf("x={x} y={y}")
    

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  GeoffR 11:45 am on 22 August 2021 Permalink | Reply

  I found one other answer, much larger than the puzzle answer, which satisfies Jim’s equation:
  255y – 245x – 16xy = 0.

  It is x = 15, y = 245.

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    Jim Randell 1:00 pm on 22 August 2021 Permalink | Reply

    And (5, 7), (15, 245) are the only (x, y) solutions in positive integers. (Although x = 7 is very close). At x ≥ 16, y becomes negative.

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  GeoffR 2:41 pm on 23 August 2021 Permalink | Reply

  % A Solution in MiniZinc
  
  var 2..10:X;  % horizontal
  var 2..50:Y;  % vertical
  
  constraint Y > X;
  
  % Number of sewn edges = (Y - 1).X + (X - 1).Y
  % Area of rectangular throw = X * Y
  % Time to make throw and sew throw edges = 
  % 6.X.Y + 5.( (Y-1).X + (X-1).Y )
  
  % Square throw unit time = 51/50 * Rectangle throw unit time
  constraint (6*X*Y + 5*( (Y - 1)*X + (X - 1)*Y ) ) * 51 * Y * Y
  == (6*Y*Y + 10*Y*(Y - 1)) * 50 * X * Y;
  
  solve satisfy;
  
  output ["Throw dimensions (ft) are X = " ++ show(X) 
  ++ ", " ++ "Y = " ++ show(Y)];
  
  
  

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