Cryptarithmetic Problem - 2

                H  O  W
             x  C  U  T
             ----------
             D  D  C  P
          D  W  P  W
       D  U  O  A   
       ----------------   
       D  C  E  P  D  P
1.Value of T + O + P ?
(a) 3(b) 8(c) 6(d) 9
2.Value of B ?
(a) 1(b) 2(c) 3(d) 4
3.Value of  2C ?
(a) 13(b) 13(c) 17(d) 11

As, W x U= _ W [ D W P W ]
For W the probablities are-

Case-1 When W={2, 4, 8} and U={6}
For W the probablities are-

Case-2 When W={5} and U={3, 7, 9}   
Taking the first scenerio
Take U=6 and problem looks like,

              H  O  W
           x  C  6  T
           ----------
           D  D  C  P
        D  W  P  W
     D  6  O  A   
     ----------------   
     D  C  E  P  D  P
Getting more progress,

              H  O  W 
           x  C  6  T
           ----------
           D  D  C  P
        D  W  P  W
     D  6  O  A   
     ----------------   
     D  C  E  P  D  P

D + 6 + (carry) = C    [Carry may be either 0, 1 or 2]
Hence, problem looks like of C= {7, 8, 9}
Start hit and trail C={7, 8, 9} and W={2, 4 ,8}

C=7 and W=2
when C=7 and W=2 then D=9 [ As, C + W = D] If we take take D=9 then,

             H  O  W
          x  C  6  T
          ----------
          D  D  C  P
       9  W  P  W
    9  6  O  A   
    ----------------   
    0  C  E  P  D  P
D=0 and D=9 two values of D this C=7 and W=2(Not true)
Now, Check the with other possible values of W and C
Let's take C=7 and W=4

then D=1 as C + W = D [last digit]
Taking D=1, C=7 and W=4 now problems looks like
              H  O  4
           x  7  6  T
           ----------
           1  1  7  P
        1  4  P  4
     1  6  O  A   
     ----------------   
     1  7  E  P  1  P
Now, you can easily predict the value of A=8  As, 7 x 4= _ A [last digit]

              H  O  4
           x  7  6  T
           ----------
           1  1  7  P
        1  4  P  4
     1  6  O  8   
     ----------------   
     1  7  E  P  1  P
Now, 
              H  O  4
                 x  7  
           ----------
           1  6  O 8
You can easily predict the value of H=2

              2  O  4
           x  7  6  T
           ----------
           1  1  7  P
        1  4  P  4
     1  6  O  8   
     ----------------   
     1  7  E  P  1  P
Now you can easily predict other values.
Value of T=5 O=3 and P=0


              2  3  4
           x  7  6  5
           ----------
           1  1  7  0
        1  4  0  4
     1  6  3  8   
     ----------------   
     1  7  9  0  1  0
H=2, O=3, W=4, C=7, U=6, T=5, D=1, S=0, E=9

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

No comments:

Post a Comment