S2T2

Jim Randell 5:26 pm on 5 July 2019 Permalink Reply
Tags: by: Graham Smithers ( 83 )   

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  Jim Randell 5:59 pm on 5 July 2019 Permalink | Reply

  The following Python program runs in 90ms.

  Run: [ @repl.it ]

  from enigma import (nsplit, subsets, irange, unpack, filter_unique, printf)
  
  # process a sequence of digits
  def process(ds):
    while True:
      n = sum(d * d for d in ds)
      if n < 10: return n
      ds = nsplit(n)
  
  # collect the results for 2 digit numbers
  ss = list((ds, process(ds)) for ds in subsets(irange(0, 9), size=2, select='R') if ds != (0, 0))
  
  # if I tell you "one of the digits is <k>" you should be able to work out the result
  for k in irange(0, 9):
    (rs, _) = filter_unique(ss, unpack(lambda ds, r: k in ds), unpack(lambda ds, r: r))
  
    # consider possible 2-digit sequences
    for (ds, _) in rs:
      # and add an extra digit
      for x in irange(0, 9):
        # the result should be the digit <k>
        if process(ds + (x,)) == k:
          printf("k={k}: {ds} + {x}")
  

  Solution: The three digits are: 5, 6, 9.

  The order of the digits in a number does not matter. 24 will have the same result as 42 (both are 2² + 4² = 20 → 2² + 0² = 4).

  The key digit is 5. Any 2-digit sequence that contains a 5 has a result of 4.

  The initial number is one of: 56, 59, 65, 95. An extra digit is added so the three digit sequence consists of the digits 5, 6, 9 (in some order). And this sequence has a result of 5.

  ---

  Most of the 2-digit numbers give a result of 4, and if the process is allowed to continue summing the squares of the digits we see the following cycle emerge:

  4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4

  Numbers that eventually result in this cycle are known as “unhappy numbers” (see [ link ]).

  The other 2-digit numbers give results of 1, 2, 5, 8, 9, we can also continue the process with these values:

  1 → 1
  2 → 4
  5 → 25 → 29 → 85 → 89 → … → 4
  8 → 64 → 52 → 29 → … → 4
  9 → 81 → 65 → 61 → 37 → … → 4

  We see that any numbers with a result of 2, 5, 8, 9 are also “unhappy”, and only those numbers that have a result of 1 are “happy”. (See OEIS A124095 [ link ]).

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  • 

    Jim Randell 3:54 pm on 7 July 2019 Permalink | Reply

    Here’s a simpler program to solve the puzzle.

    Run: [ @repl.it ]

    from enigma import (nsplit, seq_all_same, irange, subsets, printf)
    
    # process a sequence of digits
    def process(*ds):
      while True:
        n = sum(d * d for d in ds)
        if n < 10: return n
        ds = nsplit(n)
    
    # look for possible key digits
    for k in irange(0, 9):
      # must give the same result when combined with all other digits
      if seq_all_same(process(k, d) for d in irange(0, 9) if k + d > 0):
    
        # look for two additional digits to give a result of k
        for (a, b) in subsets(irange(0, 9), size=2, select='R'):
          if process(k, a, b) == k and k + a + b > 0:
            printf("k={k}: {s}", s=sorted([k, a, b]))
    

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