New Scientist Enigma 516 – The ABC of Division

by Eric Emmet

From Issue #1668, 10th June 1989

In the following division sum each letter stands for a different digit. Rewrite the sum with the letter replaced by digits

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Using Jim Randell’s Python interface for MiniZinc (available here)


from minizinc import MiniZinc, var, alphametic, substitute

# the MiniZinc model
model = f"""

include "globals.mzn";

% declare the decision variables
{var("0..9", "abcehmnqsw")};

% that are all different
constraint all_different([{','.join("abcehmnqsw")}]);

% overall division sum
constraint {alphametic("{csm} * {ween} + {shs} = {qsmcaq}")};

% partial products
constraint {alphametic("{w} * {csm} =  {csm}")};
constraint {alphametic("{e} * {csm} =  {nhn}")};
constraint {alphametic("{n} * {csm} = {sqwe}")};

% partial differences
constraint {alphametic(" {qsm} -  {csm} = {wbb}")};
constraint {alphametic("{wbbc} -  {nhn} = {wea}")};
constraint {alphametic("{weaa} -  {nhn} = {snn}")};
constraint {alphametic("{snnq} - {sqwe} = {shs}")};

solve satisfy;
"""

# the symbols used in the division sum
fmt = """
              w e e n
        -------------
  c s m ) q s m c a q
          c s m 
          -----
          w b b c 
            n h n
          -------
            w e a a 
              n h n
              -----
              s n n q 
              s q w e
              -------
                s h s
                =====
"""

p = MiniZinc(model)
for s in p.solve():
  print(p.substitute(s, fmt))

from minizinc import MiniZinc, var, alphametic, substitute

# the MiniZinc model

model = f"""

include "globals.mzn";

% declare the decision variables

{var("0..9", "abcehmnqsw")};

% that are all different

constraint all_different([{','.join("abcehmnqsw")}]);

% overall division sum

constraint {alphametic("{csm} * {ween} + {shs} = {qsmcaq}")};

% partial products

constraint {alphametic("{w} * {csm} = {csm}")};

constraint {alphametic("{e} * {csm} = {nhn}")};

constraint {alphametic("{n} * {csm} = {sqwe}")};

% partial differences

constraint {alphametic(" {qsm} - {csm} = {wbb}")};

constraint {alphametic("{wbbc} - {nhn} = {wea}")};

constraint {alphametic("{weaa} - {nhn} = {snn}")};

constraint {alphametic("{snnq} - {sqwe} = {shs}")};

solve satisfy;

"""

# the symbols used in the division sum

fmt = """

w e e n

-------------

c s m ) q s m c a q

c s m

-----

w b b c

n h n

-------

w e a a

n h n

-----

s n n q

s q w e

-------

s h s

=====

"""

p = MiniZinc(model)

for s in p.solve():

print(p.substitute(s, fmt))

New Scientist Enigma 516 – The ABC of Division

by Eric Emmet

From Issue #1668, 10th June 1989

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