alpar@9: /* MONEY, a crypto-arithmetic puzzle */
alpar@9: 
alpar@9: /* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
alpar@9: 
alpar@9: /* This is the classic example of a crypto-arithmetic puzzle published
alpar@9:    in the Strand Magazine by Henry Dudeney:
alpar@9: 
alpar@9:         S E N D
alpar@9:       +
alpar@9:         M O R E
alpar@9:       ---------
alpar@9:       M O N E Y
alpar@9: 
alpar@9:    In this puzzle the same letters mean the same digits. The question
alpar@9:    is: how to replace all the letters with the respective digits that
alpar@9:    makes the calculation correct?
alpar@9: 
alpar@9:    The solution to this puzzle is:
alpar@9:    O = 0, M = 1, Y = 2, E = 5, N = 6, D = 7, R = 8, and S = 9.
alpar@9: 
alpar@9:    References:
alpar@9:    H. E. Dudeney, in Strand Magazine vol. 68 (July 1924), pp. 97, 214.
alpar@9: 
alpar@9:    (From Wikipedia, the free encyclopedia.) */
alpar@9: 
alpar@9: set LETTERS := { 'D', 'E', 'M', 'N', 'O', 'R', 'S', 'Y' };
alpar@9: /* set of letters */
alpar@9: 
alpar@9: set DIGITS := 0..9;
alpar@9: /* set of digits */
alpar@9: 
alpar@9: var x{i in LETTERS, d in DIGITS}, binary;
alpar@9: /* x[i,d] = 1 means that letter i is digit d */
alpar@9: 
alpar@9: s.t. one{i in LETTERS}: sum{d in DIGITS} x[i,d] = 1;
alpar@9: /* each letter must correspond exactly to one digit */
alpar@9: 
alpar@9: s.t. alldiff{d in DIGITS}: sum{i in LETTERS} x[i,d] <= 1;
alpar@9: /* different letters must correspond to different digits; note that
alpar@9:    some digits may not correspond to any letters at all */
alpar@9: 
alpar@9: var dig{i in LETTERS};
alpar@9: /* dig[i] is a digit corresponding to letter i */
alpar@9: 
alpar@9: s.t. map{i in LETTERS}: dig[i] = sum{d in DIGITS} d * x[i,d];
alpar@9: 
alpar@9: var carry{1..3}, binary;
alpar@9: /* carry bits */
alpar@9: 
alpar@9: s.t. sum1: dig['D'] + dig['E']            = dig['Y'] + 10 * carry[1];
alpar@9: s.t. sum2: dig['N'] + dig['R'] + carry[1] = dig['E'] + 10 * carry[2];
alpar@9: s.t. sum3: dig['E'] + dig['O'] + carry[2] = dig['N'] + 10 * carry[3];
alpar@9: s.t. sum4: dig['S'] + dig['M'] + carry[3] = dig['O'] + 10 * dig['M'];
alpar@9: s.t. note: dig['M'] >= 1; /* M must not be 0 */
alpar@9: 
alpar@9: solve;
alpar@9: /* solve the puzzle */
alpar@9: 
alpar@9: display dig;
alpar@9: /* and display its solution */
alpar@9: 
alpar@9: end;