S2T2

Jim Randell 9:03 am on 25 February 2024 Permalink Reply
Tags: by: C Higgins ( 36 ), by: H Bradley ( 36 )   

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  Jim Randell 9:04 am on 25 February 2024 Permalink | Reply

  See also: Teaser 2926.

  The arrangement looks like this:

  

  If we consider the head, body, one arm and the enclosing circle we have four mutually tangent circles, so we can use Descartes’ Theorem [ @wikipedia ] to express a relationship between their curvatures.

  If the curvatures are represented by k (where: k = 1 / r, i.e. the curvature is the reciprocal of the corresponding radius), then we have:

  (Σk)² = 2Σ(k²)

  In this instance the circles have radii of r, 3r, r − 1, 4r and so the curvatures are:

  k1 = 1/r
  k2 = 1/3r
  k3 = 1/(r − 1)
  k4 = −1/4r

  The curvature of the fourth circle is negative as it circumscribes the other three circles.

  We can solve the puzzle numerically by simply considering possible r values.

  The following program runs in 65ms. (Internal runtime is 102µs).

  Run: [ @replit ]

  from enigma import (Rational, irange, inf, sq, printf)
  
  Q = Rational()
  
  # consider possible values for r
  for r in irange(2, inf):
  
    # calculate the 4 curvatures
    ks = (Q(1, r), Q(1, 3 * r), Q(1, r - 1), Q(-1, 4 * r))
  
    # check Descartes' Theorem
    if sq(sum(ks)) == 2 * sum(map(sq, ks)):
      printf("r = {r}")
      break
  

  Solution: The head has a radius of 13 cm.

  Or we can solve it by finding the roots of a polynomial.

  This Python program runs in 66ms. (Internal runtime is 3.2ms).

  Run: [ @replit ]

  from enigma import (Polynomial, sq, printf)
  
  # consider the numerators of the curvatures, where the denominator is 12r(r-1)
  ks = [
    12 * Polynomial([-1, 1]),  # k1 = 12(r-1) / 12r(r-1) = 1/r
     4 * Polynomial([-1, 1]),  # k2 = 4(r-1) / 12r(r-1) = 1/3r
    12 * Polynomial([ 0, 1]),  # k3 = 12r / 12r(r-1) = 1/(r-1)
    -3 * Polynomial([-1, 1]),  # k4 = -3(r-1) / 12r(r-1) = -1/4r
  ]
  
  # Descartes' Theorem: sq(sum(ks)) = 2 * sum(map(sq, ks))
  p = sq(sum(ks)) - 2 * sum(map(sq, ks))
  
  # solve the polynomial for positive integer solutions
  for r in p.rational_roots(domain='Z', include='+'):
    printf("r = {r}")
  

  Analytically, we find the polynomial we are solving is:

  r² − 26r + 169 = 0
  ⇒
  (r − 13)² = 0
  r = 13

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