Substitute each letter by a different digit from 0 to 9 to satisfy this alphametic multiplication problem, where X represents the multiplication symbol.

         M E N
X       A R T
----------------
      B C O N
   B R B E
N B B N
----------------
N A T  I O N

Note: While a solution is trivial with the aid of a computer program, can you derive it without one?

Neither A, R or T can be 0.
As the products of each multiplication is given as 4 digits, unless B=0, neither A nor R can be 1. T not = 1!
[N*T = xN], [N*A = xN] and [N*E not = xN] gives N=5, T=(7 or 9) and A=(7 or 9).
[MEN*A = NBBN] where N=5 gives A=(7,7 or 9), B=(0,7 or 4) and M=(7, 8 or 6) and E=(1, 2 or 0), respectively.
Since (T or A)=7, neither B nor M can be 7, therefore E=0, B=4, M=6, T=7 and A=9.
With MEN = 605, and BCON = 4xx5 and BRBE = 4x40, the unidentified digits result in C as 2, O as 3 and R as 8.
Finally, MEN * ART = 605 * 987 = 597135, thus, the last previously unidentified digit, I is 1.

0 1 2 3 4 5 6 7 8 9
E I C O B N M T R A


      M E N           6 0 5
x     A R T     x     9 8 7
-----------     -----------
    B C O N         4 2 3 5
  B R B E         4 8 4 0  
N B B N         5 4 4 5    
-----------     -----------
N A T I O N     5 9 7 1 3 5

Edited on August 24, 2008, 9:57 am

Posted by Dej Mar on 2008-08-23 22:39:39