S2T2

Jim Randell 8:15 am on 13 February 2026 Permalink Reply
Tags: by: Nick Jones ( 2 )   

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  Jim Randell 8:15 am on 13 February 2026 Permalink | Reply

  Suppose the cake has radius r and height h.

  Then, if n layers of chocolate, each of thickness t, are added, the total volume of chocolate used is:

  V = 𝝅 (r + n.t)^2 h − 𝝅 r^2 h

  And this same volume can also be used to make 9 solid cylinders of chocolate, each with a radius of 20 mm:

  V = 9 𝝅 20^2 h

  Hence:

  𝝅 (r + n.t)^2 h − 𝝅 r^2 h = 9 𝝅 20^2 h
  (r + n.t)^2 − r^2 = 60^2

  The LHS is the difference of 2 squares, so can be rewritten:

  (n.t)(2r + n.t) = 60^2

  So we can solve the puzzle by looking at the divisors of 60^2.

  The following Python program runs in 68ms. (Internal runtime is 96µs).

  from enigma import (divisors_pairs, div, is_prime, printf)
  
  # consider divisors of 3600
  for (a, b) in divisors_pairs(60, 2):
    r = div(b - a, 2)
    if r is None or not is_prime(r): continue
  
    printf("r={r} [nt={a}]")
    for (n, t) in divisors_pairs(a, every=1):
      if not (t % 2 == 1 and is_prime(t)): continue
      printf("-> n={n} t={t}")
  

  Solution: Willy applied 10 layers of chocolate to the cake.

  The situation is apparently:

  cake radius = 11 mm
  layer thickness = 5 mm
  number of layers = 10

  So the cake is only 22 mm across, and it was then coated with 10 layers of 5 mm chocolate. Which means the finished item was 122 mm across with only a tiny 22 mm centre consisting of cake.

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  Ruud 9:48 am on 13 February 2026 Permalink | Reply

  import peek
  import istr
  
  for r, d, n in istr.product(istr.primes(1000), istr.primes(3, 1000), range(2, 20)):
      if (r + n * d) ** 2 - r**2 == 9 * (20**2):
          peek(r, n, d)
  

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