S2T2

Jim Randell 12:33 pm on 17 July 2022 Permalink Reply
Tags: by: Graham Smithers ( 82 )   

• 

  Jim Randell 12:34 pm on 17 July 2022 Permalink | Reply

  See: [@wikipedia].

  We can use the [[ SubstitutedExpression ]] solver from the enigma.py library to solve this puzzle.

  The following run file executes in 72ms. The internal runtime of the generated program is 3.1ms).

  Run: [ @replit ]

  #! python3 -m enigma -rr
  
  SubstitutedExpression
  
  --digits="1-9"
  
  # is <n> an H-number?
  --code="H = lambda n: n % dsum(n) == 0"
  
  # the 9-digit number is: abcdefghi
  "H({abcdefghi})"
  
  # three 2-digit H-numbers and a 3-digit H-number
  "H({ab})"
  "H({cd})"
  "H({ef})"
  "H({ghi})"
  
  # 5-digit H-numbers
  "H({efghi})"
  "H({edcba})"
  
  --answer="{abcdefghi}"
  

  Solution: The 9-digit number is: 273684915.

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• 

  GeoffR 2:16 pm on 17 July 2022 Permalink | Reply

  
  % A Solution in MiniZinc 
  include "globals.mzn";
  
  var 1..9: a; var 1..9: b; var 1..9: c;
  var 1..9: d; var 1..9: e; var 1..9: f;
  var 1..9: g; var 1..9: h; var 1..9: i;
  
  constraint all_different([a,b,c,d,e,f,g,h,i]);
  
  % 2-digit Harshad numbers
  constraint (10*a + b) mod (a + b) == 0
  /\ (10*c + d) mod (c + d) == 0
  /\ (10*e + f) mod (e + f) == 0;
  
  % 3-digit Harshad number
  constraint (100*g + 10*h + i) mod (g + h + i) == 0;
  
  % 5-digit Harshad numbers
  constraint (10000*e + 1000*f + 100*g + 10*h + i) mod(e + f + g + h + i) == 0;
  constraint (10000*e + 1000*d + 100*c + 10*b + a) mod(e + d + c + b + a) == 0;
  
  % 9-digit Harshad number
  constraint (a * pow(10,8) + b * pow(10,7) + c * pow(10,6)
  + d * pow(10,5) + e * pow(10,4) + f * pow(10,3)
  + g * pow(10,2) + h * 10 + i) mod (a+b+c+d+e+f+g+h+i) == 0;
  
  solve satisfy;
  
  output ["9-figure number = " ++ " \(a)\(b)\(c)\(d)\(e)\(f)\(g)\(h)\(i)" ];
  
  % 9-figure number = 273684915
  % ----------
  % ==========
  
  

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• 

  Frits 3:40 pm on 17 July 2022 Permalink | Reply

     
  from itertools import permutations
  from functools import reduce
  
  # convert digits sequence to number
  d2n = lambda s: reduce(lambda x, y: 10 * x + y, s)
  
  # calculate H-number
  H = lambda s: d2n(s) % sum(s) == 0
  
  i = 5  # as abcdefghi is divisible by 45
  
  # ghi:   sum(g,h) must be even as sum(g,h,i) must be odd
  # efghi: sum(e,f) must be even as sum(e,f,g,h,i) must be odd
  # ef:    e and f not both odd otherwise 10*d + e is not divisible by e + f
  #        so e and f are both even
  # ab:    a and b not both odd
  # cd:    c and d not both odd
  # g an h must be both odd otherwise a, b, c and d must all be odd
  
  # ....eeoo5
  # edcba : a must be even as sum(e,d,c,b,a) is even
  # e...eeoo5
  # as c and d are not both odd b must be odd
  # eo..eeoo5
  
  for (b, g, h) in permutations([1, 3, 7, 9], 3):
    if not H((g, h, i)): continue
    for (a, e, f) in permutations([2, 4, 6, 8], 3):
      if not H((a, b)): continue
      if not H((e, f)): continue
      if not H((e, f, g, h, i)): continue
      rest = set(range(1, 10)).difference({a, b, e, f, g, h, i})
      for (c, d) in permutations(rest):
        if not H((c, d)): continue
        if not H((e, d, c, b, a)): continue
        if not H((a, b, c, d, e, f, g, h, i)): continue
        print("the 9-figure number:", d2n((a, b, c, d, e, f, g, h, i)))
  

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• 

  GeoffR 6:43 pm on 17 July 2022 Permalink | Reply

  I carried out some further analysis to see how the number of Harshad numbers reduced with different arrangements of numbers.

  1) For numbers ab, cd, ef, ghi and abcdefghi, this gave 96 possible 9-digit Harshad numbers.

  2) Adding abcde to (1) above, I found 16 possible Harshad numbers for abcdefghi:

   a b c d e f g h i    a b c d e f g h i    a b c d e f g h i
  ------------------------------------------------------------
   2 7 3 6 4 8 1 9 5,   2 7 3 6 8 4 9 1 5,   2 7 6 3 4 8 1 9 5,   
   2 7 6 3 8 4 9 1 5,   3 6 2 7 4 8 1 9 5,   3 6 2 7 8 4 9 1 5,   
   3 6 7 2 4 8 1 9 5,   3 6 7 2 8 4 9 1 5,   6 3 2 7 4 8 1 9 5,   
   6 3 2 7 8 4 9 1 5,   6 3 7 2 4 8 1 9 5,   6 3 7 2 8 4 9 1 5,   
   7 2 3 6 4 8 1 9 5,   7 2 3 6 8 4 9 1 5,   7 2 6 3 4 8 1 9 5,   
   7 2 6 3 8 4 9 1 5.
  

  3) Finally, adding number edcba to (2) above, this gave the single 9-digit answer of 273684915.
  (top row, middle value)

  Interesting to note how the 16 solutions illustrate aspects of analysis in Frits code.

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