S2T2

Jim Randell 7:40 am on 17 March 2026 Permalink Reply
Tags: by: Victor Bryant ( 143 )   

• 

  Jim Randell 7:41 am on 17 March 2026 Permalink | Reply

  I have assumed that digits do not “wrap around”, and so if “0” is typed the machine always replaces it with “1”. And if “9” is typed the machine always replaces it with “8”.

  This Python program runs in 75ms. (Internal runtime is 744µs).

  from enigma import (defaultdict, irange, subsets, cproduct, is_prime, sprintf as f, join, printf)
  
  # possible incorrect digits
  ds = {
    0: [1], 1: [0, 2], 2: [1, 3], 3: [2, 4], 4: [3, 5],
    5: [4, 6], 6: [5, 7], 7: [6, 8], 8: [7, 9], 9: [8],
  }
  
  # collect candidate inputs by possible output
  rs = defaultdict(list)
  
  # choose an input sum: a * b = xy
  for (a, b) in subsets(irange(1, 9), size=2):
    xy = a * b
    (x, y) = divmod(xy, 10)
    if x == 0: continue
  
    # can we adjust all the digits by 1 to give a result that is off by one?
    for (a_, b_, x_, y_) in cproduct(ds[k] for k in (a, b, x, y)):
      if x_ == 0: continue
      xy_ = 10*x_ + y_
      if abs(a_ * b_ - xy_) == 1 and is_prime(xy_):
        (expr, expr_) = (f("{a} * {b} = {xy}"), f("{a_} * {b_} = {xy_}"))
        rs[expr_].append(expr)
        printf("[{expr} -> {expr_}]")
  
  # look for ambiguous outputs
  for (k, vs) in rs.items():
    if len(vs) > 1:
      printf("{k} <- {vs}", vs=join(vs, sep="; "))
  

  Solution: The display shows: “5 × 6 = 31”.

  And there are two inputs that may lead to this display:

  “4 × 5 = 20” → “5 × 6 = 31” (++++)
  “6 × 7 = 42” → “5 × 6 = 31” (−−−−)

  Note that in both these cases the digits are all changed in the same direction.

  ---

  However, if we allow the digits to “wrap around” so that “0” could be replaced by “1” or “9”, and “9” could be replaced by “0” or “8”, then there are further possible displays that have ambiguous origins:

  “3 × 6 = 18” → “4 × 7 = 29” (++++)
  “5 × 6 = 30” → “4 × 7 = 29” (−+−−)

  “4 × 5 = 20” → “3 × 6 = 19” (−+−−)
  “4 × 7 = 28” → “3 × 6 = 19” (−−−+)

  And these require that some digits in the expression are moved up and some moved down.

  LikeLike