S2T2

Jim Randell 3:43 pm on 23 July 2020 Permalink Reply
Tags: by: Danny Roth ( 84 )   

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  Jim Randell 3:43 pm on 23 July 2020 Permalink | Reply

  If the combination can be considered as three 2-digit primes, then it cannot contain 0 (as a 2-digit prime cannot begin or end with 0).

  This Python program looks for 6 different (non-zero) digits that have a square for both their sum and product, and looks for 6-digit multiples of the difference between these that use the same digits, and can be split into three 2-digit prime numbers. It runs in 52ms.

  Run: [ @replit ]

  from enigma import (subsets, irange, multiply, is_square, divc, is_prime, nsplit, printf)
  
  # look for 6 different digits
  for ds in subsets(irange(1, 9), size=6, select="C"):
    # the sum and product must both be squares
    s = sum(ds)
    p = multiply(ds)
    if not (is_square(s) and is_square(p)): continue
    (d, ds) = (p - s, set(ds))
    # consider 6-digit multiples of d
    for n in irange(d * divc(100000, d), 999999, step=d):
      # does it use the same digits?
      if ds.difference(nsplit(n)): continue
      # does it split into primes?
      if not all(is_prime(p) for p in nsplit(n, base=100)): continue
      # output solution
      printf("{n} [s={s} p={p} d={d}]")
  

  Solution: The combination is 296143.

  There is only one set of 6 different digits that have a square for both their sum and product: 1, 2, 3, 4, 6, 9 (sum = 5², product = 36²). So the combination is a rearrangement of these digits that is a multiple of 1271, that can be split into three 2-digit primes. In fact 296143 is the only multiple of 1271 that uses the required digits, so the prime test is superfluous.

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  GeoffR 8:38 pm on 23 July 2020 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; 
  
  constraint all_different([A,B,C,D,E,F]);
  
  % three two-figure primes
  var 11..97: AB = 10*A + B;
  var 11..97: CD = 10*C + D;
  var 11..97: EF = 10*E + F;
  
  var 720..60480: dig_prod = A * B * C * D * E * F;
  var 21..39: dig_sum = A + B + C + D + E + F;
  
  % Safe combination is ABCDEF
  var 100000..999999: comb = 100000*A + 10000*B + 1000*C
  + 100*D + 10*E + F;
  
  predicate is_prime(var int: x) = 
  x > 1 /\ forall(i in 2..1 + ceil(sqrt(int2float(ub(x))))) 
  ((i < x) -> (x mod i > 0));
  
  predicate is_sq(var int: y) =
    let {
       var 1..ceil(sqrt(int2float(ub(y)))): z
    } in
     z*z = y;
  
  % the combination is split into three 2-digit primes
  constraint is_prime(AB) /\ is_prime(CD) /\ is_prime(CD);
  
  % digit product and digit sum are both squares
  constraint is_sq(A * B * C * D * E * F);
  constraint is_sq(A + B + C + D + E + F);
  
  % the six-figure combination was actually a multiple of 
  % the difference between that product and sum.
  constraint comb mod (dig_prod - dig_sum) == 0;
  
  solve satisfy;
  
  output["Safe combination is " ++ show(comb)
  ++ "\nDigits product = " ++ show(dig_prod)];
  
  % Safe combination is 296143
  % Digits product = 1296
  % ----------
  % ==========
  
  
  

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  GeoffR 4:44 pm on 24 July 2020 Permalink | Reply

  I carried out some further tests to see if there were any constraints which were not essential to get the correct answer.

  I found any one of the following MiniZinc constraints could be removed, and the single correct answer obtained:

  1) requirement for digits to be all different
  2) requirement for three 2-digit primes
  3) requirement for sum of digits to be a square
  4) requirement product of digits to be a square.

  However, I could not remove more than one of these constraints and get the correct answer.

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