S2T2

Jim Randell 11:09 am on 10 October 2025 Permalink Reply
Tags: by: Des MacHale ( 12 )   

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  Jim Randell 11:10 am on 10 October 2025 Permalink | Reply

  See also: Puzzle #64.

  We are looking for solutions to the equation:

  R² = r² + 10²

  Here is a solution using the pells.py library (a bit of overkill perhaps for a simple problem, but the code is already written, so here goes):

  from enigma import printf
  import pells
  
  # solve: R^2 - r^2 = 10^2
  for (R, r) in pells.diop_quad(1, -1, 100):
    if R > r > 0:
      printf("R={R} r={r}")
  

  Solution: The circles have radii of 24 m and 26 m.

  ---

  Manually, we solve:

  R² − r² = 100
  (R − r)(R + r) = 100

  for positive integer values of R, r.

  So we can consider divisor pairs of 100:

  100 = 1×100 ⇒ R = 50.5, r = 49.5
  100 = 2×50 ⇒ R = 26, r = 24 [SOLUTION]
  100 = 4×25 ⇒ R = 14.5, r = 10.5
  100 = 5×20 ⇒ R = 12.5, r = 7.5
  100 = 10×10 ⇒ R = 10; r = 0

  Only one of the pairs gives positive integer solutions.

  And this is the same process followed by the program.

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  Ruud 8:33 am on 11 October 2025 Permalink | Reply

  Strictly speaking the circles could also have a radius of 0 and 10.

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  Ruud 8:37 am on 11 October 2025 Permalink | Reply

  for r in range(50):
      for R in range(r, 50):
          if R * R == 100 + r * r:
              print(r, R)
  

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