/* CRYPTO, a crypto-arithmetic puzzle */

 /* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */

 /* This problem comes from the newsgroup rec.puzzle.

    The numbers from 1 to 26 are assigned to the letters of the alphabet.

    The numbers beside each word are the total of the values assigned to

    the letters in the word (e.g. for LYRE: L, Y, R, E might be to equal

    5, 9, 20 and 13, or any other combination that add up to 47).

    Find the value of each letter under the equations:

    BALLET  45     GLEE  66     POLKA      59     SONG     61

    CELLO   43     JAZZ  58     QUARTET    50     SOPRANO  82

    CONCERT 74     LYRE  47     SAXOPHONE 134     THEME    72

    FLUTE   30     OBOE  53     SCALE      51     VIOLIN  100

    FUGUE   50     OPERA 65     SOLO       37     WALTZ    34

    Solution:

    A, B,C, D, E,F, G, H, I, J, K,L,M, N, O, P,Q, R, S,T,U, V,W, X, Y, Z

    5,13,9,16,20,4,24,21,25,17,23,2,8,12,10,19,7,11,15,3,1,26,6,22,14,18

    Reference:

    Koalog Constraint Solver <http://www.koalog.com/php/jcs.php>,

    Simple problems, the crypto-arithmetic puzzle ALPHACIPHER. */

 set LETTERS :=

 {     'A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M',

       'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V', 'W', 'X', 'Y', 'Z'

 };

 /* set of letters */

 set VALUES := 1..card(LETTERS);

 /* set of values assigned to the letters */

 set WORDS;

 /* set of words */

 param total{word in WORDS};

 /* total[word] is the total of the values assigned to the letters in

    the word */

 var x{i in LETTERS, j in VALUES}, binary;

 /* x[i,j] = 1 means that letter i is assigned value j */

 s.t. phi{i in LETTERS}: sum{j in VALUES} x[i,j] = 1;

 s.t. psi{j in VALUES}: sum{i in LETTERS} x[i,j] = 1;

 s.t. eqn{word in WORDS}: sum{k in 1..length(word), j in VALUES}

       j * x[substr(word,k,1), j] = total[word];

 solve;

 printf{i in LETTERS} "  %s", i;

 printf "\n";

 printf{i in LETTERS} " %2d", sum{j in VALUES} j * x[i,j];

 printf "\n";

 data;

 param :  WORDS :   total :=

          BALLET       45

          CELLO        43

          CONCERT      74

          FLUTE        30

          FUGUE        50

          GLEE         66

          JAZZ         58

          LYRE         47

          OBOE         53

          OPERA        65

          POLKA        59

          QUARTET      50

          SAXOPHONE   134

          SCALE        51

          SOLO         37

          SONG         61

          SOPRANO      82

          THEME        72

          VIOLIN      100

          WALTZ        34 ;

 end;