S2T2

Jim Randell 4:36 pm on 11 June 2021 Permalink Reply
Tags: by: Victor Bryant ( 142 )   

• 

  Jim Randell 4:50 pm on 11 June 2021 Permalink | Reply

  The following Python program runs in 52ms.

  Run: [ @replit ]

  from enigma import irange, tri, div, printf
  
  # collect rectangles with sides up to x, and total area
  (rs, t) = (list(), 0)
  for x in irange(1, 49):
    rs.extend((a, x) for a in irange(1, x))
    t += x * tri(x)
  
    # calculate the number of tiles and the total area
    n = len(rs)
    # look for an integer mean
    m = div(t, n)
    if m is None: continue
  
    # look for tiles with the mean area
    ts = list((a, b) for (a, b) in rs if a * b == m)
    if len(ts) > 1:
      printf("x={x} [m={m}; ts={ts}]")
  

  or (shorter, but doesn’t track tile dimensions):

  from enigma import multiset, irange, div, printf
  
  # collect rectangles with sides up to x
  rs = multiset()
  for x in irange(1, 49):
    rs.update(a * x for a in irange(1, x))
    # calculate integer mean tile area
    m = div(rs.sum(), rs.size())
    # look for tiles with the mean area
    if rs.count(m) > 1:
      printf("x={x} [m={m}]")
  

  Solution: The artist is 37.

  So, there were originally 703 tiles with a total area of 255892 cm², given a mean area of 364 cm².

  There are 2 tiles with this area (= 14×26, 13×28), and the removal of either (or both) will leave the mean area unchanged.

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  • 

    Jim Randell 8:47 am on 12 June 2021 Permalink | Reply

    And here’s a version that doesn’t collect the rectangles at all (and does report tile dimensions):

    The number of tiles is given by the triangular numbers:

    T(n) = n (n + 1) / 2

    And the total area is given by the 4-dimensional pyramidal numbers (see: OEIS A001296):

    A(n) = n (n + 1) (n + 2) (3n + 1) / 24

    And hence the mean area of a tile is:

    M(n) = A(n) / T(n) = (n + 2) (3n + 1) / 12

    So we can look for an integer mean, that has (at least) 2 divisor pairs in the required range:

    from enigma import irange, div, divisors_pairs, printf
    
    # consider ages
    for x in irange(1, 49):
      # calculate integer mean
      m = div((x + 2) * (3 * x + 1), 12)
      if m is None: continue
      # find tiles with mean area
      ts = list((a, b) for (a, b) in divisors_pairs(m) if b <= x)
      if len(ts) > 1:
        printf("x={x} [m={m}; ts={ts}]")
    

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    • 

      Frits 10:05 am on 12 June 2021 Permalink | Reply

      Nice.

      Proof:

      see [ https://www.wolframalpha.com/input/?i=%28sum+x*x*%28x%2B1%29%2F2%2C+x%3D1+to+n%29+ ]

      total area = sum(y=1..x) 1/2 y y (y + 1) = 1/24 x (x + 1) (x + 2) (3x + 1)
      number of tiles = x (x+1) / 2
      integer mean = (x + 2) * (3 * x + 1) / 12

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    • 

      Tony Brooke-Taylor 8:07 am on 14 June 2021 Permalink | Reply

      To get an integer mean, x must satisfy the requirement that (x+2).(3x+1) mod 12 = 0. This requires that x be equal to either 1+12i or 10+12i (i=0,1,…).

      This plot gives a visual representation of that requirement.

      
      import numpy as np
      import matplotlib.pyplot as plt
      
      x = np.asarray([a for a in range(1,51)])
      y = np.asarray([(a+2)*(3*a+1)%12 for a in range(1,51)])
      
      plt.scatter(x, y)
      plt.title('y=(x + 2).(3x + 1) mod 12')
      plt.xlabel('x')
      plt.ylabel('y')
      plt.show()
      
      

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