Substitute each letter by a different decimal digit from 0 to 9 to satisfy this alphametic division puzzle.

                     A N Y
        -------------------
I T S |  I S L  A N D
           T N N A
        -------------------
              T D U N
              T  L E N
        ------------------
                 S Y U D
                 S T N Y
         -------------------
                    S T I

Note: Derive a non computer-assisted solution.

Given that A*S = xA, N*S = xN, and Y*S = xS such that x represents the 10¹-digit , S must be equal to 1 (thus, no 10¹-digit resulting in the product.

Only two products involving the multiplication of the distinct digits ITS with A, such that S = 1, will result in the pattern TNNA. These are where (A, I, T) = (9, 3, 2) and (4, 7, 2). Thus, T = 2; A = (4 or 9); and, correspondingly, I = (7 or 3).

With ITS * N and the deduced possible values of A, I and T and the value of S, the only value that results in the distinct digit pattern TLEN is 2568, with N = 8 and ITS = 321. Thus,
(A, E, I, L, N, S, T) = (4, 6, 3, 5, 8, 1, 2) or (9, 6, 3, 5, 8, 1, 2).

Now with the values known of more of the letter represented digits, the only value possible for 321 * Y is where Y = 4, resulting in the product 1284. Thus, as each letter represents a distinct digit, A = 9 and
(1, 2, 3, 4, 5, 6, 8, 9) = (S, T, I, Y, L, E, N, A).

ISLA - TNNA = 3159 – 2889 = 270 = TDU.
Therefore, D = 7 and U = 0.
Thus, (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) = (U, S, T, I, Y, L, E, D, N, A)

              9 8 4
      ---------------
3 2 1 | 3 1 5 9 8 7
        2 8 8 9
      ---------------
          2 7 0 8
          2 5 6 8
      ---------------
            1 4 0 7
            1 2 8 4
      ---------------
              1 2 3

Posted by Dej Mar on 2008-11-19 10:00:23