S2T2

Jim Randell 11:52 am on 13 June 2023 Permalink Reply
Tags: by: Adrian Somerfield ( 13 )   

• 

  Jim Randell 11:52 am on 13 June 2023 Permalink | Reply

  See also: Enigma 1358.

  This Python program collects numerical values of “1101” in various bases, such that the value is not more than 4 digits in decimal.

  It then looks for three of these values that sum to a fourth, and applies the remaining conditions.

  It runs in 56ms. (Internal runtime is 518µs).

  Run: [ @replit ]

  from enigma import (
    irange, inf, base2int, subsets, icount_exactly,
    is_prime, seq_all_different, join, printf
  )
  
  # is x divisible by 3?
  div3 = lambda x: x % 3 == 0
  
  # find values for <s> in various bases
  d = dict()
  for b in irange(2, inf):
    n = base2int('1101', base=b)
    if n > 9999: break
    d[n] = b
  
  # look for 3 of the values that sum to a 4th
  for ns in subsets(d.keys(), size=3, fn=list):
    t = sum(ns)
    if t < 1000 or t not in d: continue
    # the last digits of the summands (in base 10) are all different
    if not seq_all_different(n % 10 for n in ns): continue
    ns.append(t)
    # exactly one of the 4 numbers is divisible by 3
    if not icount_exactly(ns, div3, 1): continue
    # exactly one of the 4 numbers is prime
    if not icount_exactly(ns, is_prime, 1): continue
    # output solution
    bs = list(d[n] for n in ns)
    printf("{ns} = {t} [bases = {bs}]",
      ns=join(ns[:-1], sep=" + "),
      bs=join(bs, sep=", ", enc="()"),
    )
  

  Solution: The result of the sum is 7221 (in decimal).

  The sum is:

  1101 [base 6] = 253 [decimal] +
  1101 [base 9] = 811 [decimal] +
  1101 [base 18] = 6157 [decimal] =
  1101 [base 19] = 7221 [decimal]

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  • 

    Frits 12:01 pm on 14 June 2023 Permalink | Reply

    @Jim, you don’t seem to have a proper check on “the total is still a four-figure number”.

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    • 

      Jim Randell 12:12 pm on 14 June 2023 Permalink | Reply

      @Frits: Oops. I’ve added in a check for that at line 19.

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• 

  GeoffR 10:21 pm on 13 June 2023 Permalink | Reply

  
  from enigma import is_prime, all_different
  from itertools import combinations
  
  # max base for 4 digits is 21
  # sum is n1 + n2 + n3 = n4
  for b1, b2, b3, b4 in combinations(range(2, 22),4):
    n4 = int('1101', base = b4)
    # the total n4 is a four-figure number
    if n4 < 10000:
      n1 = int('1101', base = b1)
      n2 = int('1101', base = b2)
      n3 = int('1101', base = b3)
      # one of the four numbers is prime
      if sum  ((is_prime(n1), is_prime(n2), is_prime(n3),\
                is_prime(n4))) == 1:
        # last digits of the three summands are all different
        if all_different( str(n1)[:-1], str(n2)[:-1], \
                          str(n3)[:-1]):
          # one of the four numbers is divisible by three
          if sum((n1 % 3 == 0, n2 % 3 == 0, n3 % 3 == 0, \
                  n4 % 3 == 0))== 1:
            # check total of sum is correct
            if n4 == n1 + n2 + n3:
              print(f"Sum is : {n1} + {n2} + {n3} = {n4}.")
              print(f"The bases are {b1}, {b2}, {b3} and {b4}.")
  
  # Sum is : 253 + 811 + 6157 = 7221.
  # The bases are 6, 9, 18 and 19.
  
  

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• 

  GeoffR 8:38 am on 14 June 2023 Permalink | Reply

  Quite slow in MiniZinc – nearly 3 sec with the Chuffed Solver.

  % A Solution in MiniZinc
  include "globals.mzn";
  
  predicate is_prime(var int: x) = 
  x > 1 /\ forall(i in 2..1 + ceil(sqrt(int2float(ub(x))))) 
  ((i < x) -> (x mod i > 0));
  
  % Four numbers with four different number bases
  % min LB = 8 + 4 + 0 + 1 = 13
  % max UB = 21**3 + 21**2 + 0 + 1 = 9703
  var 13..9703:N1; var 13..9703:N2; var 13..9703:N3; var 13..9703:N4;
  var 2..21:b1; var 2..21:b2; var 2..21:b3; var 2..21:b4;
   
  % put bases in order
  constraint b2 > b1 /\ b3 > b2 /\ b4 > b3;
  
  % Form 4 numbers in 4 different bases in format 1101
  constraint N1 == 1 * b1 * b1 * b1 + 1 * b1 * b1 + 0 + 1;
  constraint N2 == 1 * b2 * b2 * b2 + 1 * b2 * b2 + 0 + 1;
  constraint N3 == 1 * b3 * b3 * b3 + 1 * b3 * b3 + 0 + 1;
  constraint N4 == 1 * b4 * b4 * b4 + 1 * b4 * b4 + 0 + 1;
  
  % Main sum
  constraint N1 + N2 + N3 == N4;
  
  % One of the four numbers is prime
  constraint sum([is_prime(N1), is_prime(N2), is_prime(N3), is_prime(N4)]) == 1;
  
  % One of the four numbers is prime is exactly divisible by three
  constraint sum([N1 mod 3 == 0, N2 mod 3 == 0, N3 mod 3 == 0, N4 mod 3 == 0]) == 1;
  
  % the last digits of the three summands are different
  constraint all_different([N1 mod 10, N2 mod 10, N3 mod 10]);
  
  solve satisfy;
  
  output["Total is " ++ show(N4) ++ "\n" 
  ++ "Sum is " ++ show(N1) ++ " + " ++ show(N2) ++ " + " ++ show(N3) ++ " = " ++ show(N4)
  ++ "\n" ++ "Bases are " ++ show([b1, b2, b3, b4])];
  
  % Total is 7221
  % Sum is 253 + 811 + 6157 = 7221
  % Bases are [6, 9, 18, 19]
  % ----------
  % ========== 
  
  
  

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