alphamet.gms : Alphametics - a Mathematical Puzzle

Description

Alphametics are a type of mathematical puzzle. Given a set of
words, written down as a long-hand addition, find the mapping
between letters and digits (values). For example:

    G E O R G I A             8 4 6 5 8 0 2
      O R E G O N     ->        6 5 4 8 6 3
    V E R M O N T             1 4 5 9 6 3 7
    -------------           ---------------
  V I R G I N I A           1 0 5 8 0 3 0 2

Additional information can be found at:

http://www.gams.com/modlib/adddocs/alphametdoc.htm


References

  • Kalvelagen, E, Model Building with GAMS. forthcoming
  • de Wetering, A V, private communication.

Large Model of Type : MIP


Category : GAMS Model library


Main file : alphamet.gms

$title Alphametics - A Mathematical Puzzle (ALPHAMET,SEQ=170)

$onText
Alphametics are a type of mathematical puzzle. Given a set of
words, written down as a long-hand addition, find the mapping
between letters and digits (values). For example:

    G E O R G I A             8 4 6 5 8 0 2
      O R E G O N     ->        6 5 4 8 6 3
    V E R M O N T             1 4 5 9 6 3 7
    -------------           ---------------
  V I R G I N I A           1 0 5 8 0 3 0 2

Additional information can be found at:

http://www.gams.com/modlib/adddocs/alphametdoc.htm


Brooke, A, Kendrick, D, and Meeraus, A, GAMS: A User's Guide. The
Scientific Press, Redwood City, California, 1988.

Keywords: mixed integer linear programming, alphametics
$offText

Set i 'letters' / g, e, o, r, i, a, n, v, m, t /;

Alias (i,j);

abort$(card(i) <> 10) "set i should contain 10 letters";

Set
   lead(i)    / g, o, v /
   k 'slices' / 1*8     /;

Parameter
   lhs(k,i) 'input count for each letter in slice'
            / 1.(a,n,t)  1
              2.(i,o,n)  1
              3.g        2 , 3.o  1
              4.(r,e,m)  1
              5.o        1 , 5.r  2
              6.e        2 , 6.o  1
              7.(g,v)    1          /
   rhs(i,k) 'result'
            / (v.8,i.7,r.6,g.5,i.4,n.3,i.2,a.1) 1.0 /;

Variable
   z      'sum of carries - some objective'
   x(i,j) 'assignment of digits or values to letters'
   y(i)   'assigned value'
   c(k)   'carry';

Binary  Variable x;

Integer Variable c;

Equation
   obj     'some objective'
   eq(k)   'defines addition for each slice'
   ydef(i) 'assigned value'
   x1(i)   'assignment constraint one'
   x2(j)   'assignment constraint two'
   ld(i)   'bound on lead letters';

obj..      z =e= sum(k, c(k - 1));

eq(k)..    c(k-1) + sum(i, lhs(k,i)*y(i)) =e= sum(i, rhs(i,k)*y(i)) + 10*c(k)$(ord(k) <> card(k));

ydef(i)..  y(i) =e= sum(j, (ord(j) - 1)*x(i,j));

x1(i)..    sum(j, x(i,j)) =e= 1;

x2(j)..    sum(i, x(i,j)) =e= 1;

ld(lead).. y(lead) =g= 1;

Model m / all /;

option optCr = 1, optCa = 100;

solve m using mip minimizing z;