S2T2

Jim Randell 9:10 am on 13 July 2023 Permalink Reply
Tags: by: Harold Reiter ( 5 )   

• 

  Jim Randell 9:11 am on 13 July 2023 Permalink | Reply

  This Python program runs in 94ms. Internal runtime is 30ms.

  Run: [ @replit ]

  from enigma import (Accumulator, irange, subsets, fraction, printf)
  
  # available digits
  digits = set(irange(1, 9))
  
  # find maximum configurations
  r = Accumulator(fn=max, collect=1)
  
  # choose the denominators R, T, V, X in increasing order
  for (R, T, V, X) in subsets(digits, size=4, select='C'):
    # allocate the numerators
    for (Q, S, U, W, Y) in subsets(digits.difference((R, T, V, X)), size=5, select='P'):
      # evaluate the expression
      (a, b) = fraction(Q, R,  S, T,  U, V,  W, X,  -Y, 1)
      if b == 1:
        r.accumulate_data(a, (Q, R, S, T, U, V, W, X, Y))
  
  # output solution(s)
  for (Q, R, S, T, U, V, W, X, Y) in r.data:
    printf("{Q}/{R} + {S}/{T} + {U}/{V} + {W}/{X} - {Y} = {r.value}")
  

  Solution: The largest possible answer is 12.

  9/1 + 7/2 + 8/3 + 5/6 − 4 = 12

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• 

  Frits 1:11 pm on 13 July 2023 Permalink | Reply

  #! python3 -m enigma -r
  
  SubstitutedExpression
  
  # Q/R + S/T + U/V + W/X - Y = AB
  "div(Q*T*V*X + R*S*V*X + R*T*U*X + R*T*V*W - R*T*V*X*Y, R*T*V*X) = AB"
  
  --answer="AB"
  --distinct="QRSTUVWXY"
  --accumulate=max
  --invalid="0,QRSTUVWXY"
  --verbose=16   
  

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  • 

    Frits 1:20 pm on 13 July 2023 Permalink | Reply

    @Jim, is it possible to get a button like crayon on PuzzlingInPython?
    It makes it less error prone to post replies.

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  • 

    Jim Randell 2:02 pm on 13 July 2023 Permalink | Reply

    Requiring the denominators be ordered makes this approach run about 10× faster:

    Run: [ @replit ]

    #! python3 -m enigma -rr
    
    SubstitutedExpression
    
    --distinct="QRSTUVWXY"
    --invalid="0,QRSTUVWXY"
    --code="q = Rational()"
    
    "as_int(q(Q, R) + q(S, T) + q(U, V) + q(W, X) - Y) = AB"
    
    "R < T" "T < V" "V < X"
    
    --answer="AB"
    --accumulate="max"
    

    (Is “crayon” a WordPress plugin? I don’t think it is possible to install plugins on a free WordPress site. It looks like you need to pay £20/month to be able to do that).

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  • 

    Frits 5:48 pm on 13 July 2023 Permalink | Reply

    Not using the fact that 5 and 7 can’t be used as denominators.

       
    from fractions import Fraction as RF
    
    # Q/R + S/T + U/V + W/X - Y = mx
    
    # available digits
    digits = set(range(1, 10))
    
    # maximum sum for <k> fractions choosing numbers from a sorted sequence <s> 
    def maxleft(s, k=4):
      assert len(s) >= 2 * k
      return int(sum([RF(s[-1 - x], s[x]) for x in range(k)]))
    
    # check if we can make total m of <k> fractions from digits in set <s> 
    def check(s, m, k=4, ns=[]):
      if k == 1:
        ls = list(s)
    
        if m in {RF(ls[0], ls[1]), RF(ls[1], ls[0])}:
          print("answer:", sum(RF(x[0], x[1]) for x in ns) + m - Y)
          print(f"we can make remaining fraction {m} with digits {s}, other "
                f"fractions = {', '.join(str(x) + '/' + str(y) for x, y in ns)}")
          exit(0)
      else:
        # start with lowest denominators
        for d in (ss := sorted(s)):
          # ascending denominators
          if ns and d < ns[-1][1]: continue
          # start with highest numerators
          for n in ss[::-1]:
            if n != d:
              yield from check(s.difference([n, d]), m - RF(n, d),
                               k - 1, ns + [(n, d)])
    
    # build a descending list of maxima per Y
    M = sorted([(maxleft(sorted(digits.difference([Y]))) - Y, Y) 
                for Y in range(1, 10)], reverse=1)
    
    inc = 0
    # keep checking until solution gets too low (-9)
    while True:
      cnt = 0
      # check Y's with the highest maximum first
      for (mx, Y) in M:
        if mx + Y - inc >= -8:
          cnt += 1
          # check if we can make total from eight remaining digits
          for c in check(digits.difference([Y]), mx + Y - inc):
            pass
            
      if not cnt: break     
      inc += 1  
    

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• 

  GeoffR 2:05 pm on 14 July 2023 Permalink | Reply

  I found two integer solutions for the equation for the largest whole number. The largest of the two integer solutions was 12.

  
  % A Solution in MiniZinc
  include "globals.mzn";
  
  var 1..9:Q; var 1..9:R; var 1..9:S; var 1..9:T;
  var 1..9:U; var 1..9:V; var 1..9:W; var 1..9:X;
  var 1..9:Y; 
  
  % Assumed LB for the sum is 1/9 + 2/8 + 3/7 + 4/6 - 5 = -3.54 (i.e  -4)
  % Assumed UB for the sum is 9/1 + 8/2 + 7/3 + 6/4 - 5 = 11.83 (i.e. 12)
  var -4..12:Z;   % Z = Q/R + S/T + U/V + W/X - Y
  
  constraint all_different( [Q, R, S, T, U, V, W, X, Y] );
  
  % Equation in integers only (for Z == Q/R + S/T + U/V + W/X - Y) is:
  constraint Z*R*T*V*X == Q*T*V*X + S*R*V*X + U*R*T*X + W*R*T*V - Y*R*T*V*X;
  
  solve maximize(Z);
  
  output [" Z = " ++ show(Z) ++ "\n" 
  ++ "[Q,R,S,T,U,V,W,X,Y,Z] = " ++ show( [Q,R,S,T,U,V,W,X,Y,Z] ) ];
  
  % Z = 11
  % [Q,R,S,T,U,V,W,X,Y,Z] = [6, 4, 9, 3, 7, 2, 8, 1, 5, 11]
  % ----------
  %  Z = 12
  % [Q,R,S,T,U,V,W,X,Y,Z] = [5, 6, 8, 3, 7, 2, 9, 1, 4, 12]
  % ----------
  % ==========
  
  

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  • 

    Frits 3:10 pm on 14 July 2023 Permalink | Reply

    Highest rational number seems to be 12.95 (9/1 + 8/2 + 7/4 + 6/5 – 3) and
    -7.5381 (1/5 + 2/6 + 3/7 + 4/8 – 9) for the lowest.

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