S2T2

Jim Randell 9:01 am on 20 May 2026 Permalink Reply
Tags: by: W S Graff-Baker ( 3 ), flawed ( 57 )   

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  Jim Randell 9:02 am on 20 May 2026 Permalink | Reply

  We place the corners of the park at (±52, ±52), and the circle is centred on (0, 0) with a radius of 19.

  If the point on the circumference of the circle is (x, y), and the distance to the nearest corner (+52, +52) is a, and the distance to the furthest corner (−52, −52) is b, then we have the following equations:

  x² + y² = 19
  (52 − x)² + (52 − y)² = a²
  (52 + x)² + (52 + y)² = b²

  And we can eliminate x and y to get:

  a² + b² = 4 × 52² + 2 × 19² = 11538

  So we can look for two squares that sum to give the required value.

  The following Python program runs in 70ms. (Internal runtime is 51µs).

  from enigma import (
    sq, sum_of_squares, circle_intersect_circle, peek, point_dist, printf
  )
  
  S = 4 * sq(52) + 2 * sq(19)
  for (a, b) in sum_of_squares(S, 2):
    printf("a={a} b={b} [S={S}]")
  
    # find one of the intersection points
    p = peek(circle_intersect_circle(((52, 52), a), ((-52, -52), b)), default=None)
    if p is None: continue
    printf("-> p={p}")
  
    # determine distance to the corners
    for q in [(52, 52), (-52, -52), (-52, 52), (52, -52)]:
      d = point_dist(p, q)
      printf("-> dist to {q} = {d:.6f}")
    printf()
  

  Solution: The distances are 63 m and 87 m.

  However there is a problem with this answer.

  If we look at the distances to each of the corners, we find that the integer distances are not the distances to the nearest and furthest corners (at least not the nearest and furthest from (x, y)).

  

  (The integer distances are shown in red).

  The distances are:

  AX = 63 m
  BX = 88.92 m
  CX = 87 m
  DX = 60.26 m

  The shortest distance is DX and the longest distance is BX.

  But the puzzle can be saved if the radius of the circle is changed from 19 m to 18 m, then the integer distances are 58 m and 90 m, and these are the nearest and furthest distances:

  

  In this case the distances are:

  AX = 58 m
  BX = 83.16 m
  CX = 90 m
  DX = 67.44 m

  In fact for N = 52 there are solutions when R = 18, 27, 48.

  And there are solutions with integer distances that are not the nearest/furthest distances when R = 19, 26, 33, 39, 51.

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