S2T2

Jim Randell 8:36 am on 10 June 2026 Permalink Reply
Tags: by: Nick MacKinnon ( 60 )   

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  Jim Randell 8:38 am on 10 June 2026 Permalink | Reply

  (See also: Teaser 1885, also set by Nick MacKinnon).

  If we denote the angles at the vertices of the star as a, b, c, d, e, and the opposite angles in the pentagon as a′, b′, c′, d′, e′, then from the triangles formed by the star we have:

  a′ + b + e = 180°
  b′ + c + a = 180°
  c′ + d + b = 180°
  d′ + e + c = 180°
  e′ + a + d = 180°

  And the internal angles of the pentagon sum to 540°, hence:

  (a′ + b′ + c′ + d′ + e′) + 2(a + b + c + d + e) = 900°
  ⇒
  a + b + c + d + e = 180°

  And these form an arithmetic progression, so we can write:

  (a, b, c, d, e) = (c − 2x, c − x, c, c + x, c + 2x)
  ⇒
  c = 36
  x ≤ 17

  If x is even then all the angles involved will also be even, so we are not going to get 4 different prime numbers among them. So we can skip even x.

  This program considers possible x values to find a viable angles for the internal vertices of the star and for the smaller pentagon.

  It runs in 74ms. (Internal runtime is 91µs).

  from enigma import (irange, icount, is_prime, printf)
  
  # consider possible common differences in the arithmetic progression
  for x in irange(1, 17, step=2):
    # construct the progression
    ns = list(irange(36 - 2*x, 36 + 2*x, step=x))
    # count the number of primes involved
    pn = icount(ns, is_prime)
  
    # calculate the internal angles of the pentagon
    (a, b, c, d, e) = ns
    Ns = (
      180 - (b + e),
      180 - (c + a),
      180 - (d + b),
      180 - (e + c),
      180 - (a + d),
    )
    # count the primes in these
    pN = icount(Ns, is_prime)
  
    # check for exactly 4 prime angles
    if pn + pN == 4:
      # output solution
      printf("a={a} x={x} -> ns={ns} ({pn} prime); Ns={Ns} ({pN} prime)")
  

  Solution: The primes are: 31, 41, 103, 113.

  The angles at the vertices of the star form the following progression: (26°, 31°, 36°, 41°, 46°).

  And the corresponding angles at the vertices of the pentagon are: (103°, 118°, 108°, 98°, 113°).

  Here is a possible construction of the star:

  

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  Ruud 1:31 pm on 10 June 2026 Permalink | Reply

  import peek
  import istr
  
  for start in range(2, 35, 2):
      inc = (180 - 5 * start) // 10
      outer_angles = [start + i * inc for i in range(5)]
      inner_angles = [180 - angle1 - angle2 for angle1, angle2 in zip(outer_angles, outer_angles[2:] + outer_angles[:2])]
      if len(prime_angles := [angle for angle in outer_angles + inner_angles if istr.is_prime(angle)]) == 4:
          peek(prime_angles)
  

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