Even Odd Cryptarithm (W. Fitch Cheney)

  O E E
    E E
------- *
E O E E
E O E
-------
O O E E

   All even digits are replaced with E and all odd with O.
   What was the multiplication?


Solution:

  A B C
    D E
------- *
F G H I
K L M
-------
N P Q R

(I)    A B C  *      E  =  F G H I
(II)   A B C  *      D  =    K L M
(III)  A B C  *    D E  =  N P Q R
(IV)   F G H  +  K L M  =    N P Q

2 <= D  as D is even.
(II)  A * D <= K <= 9
Therefore 2*A*D <= 9*D and so A <= 9/2.
As A is odd we have A is 1 or 3.
Asume A=1 than A B C < 200. Therefore the product (I) 
is lower than 2000. But F=1 contradicts the condition F is even.
So we get A=3. Now looking at (II) again we have 3 * D <= 9.
So we get D=2 as D is even.
(I)  A B C  *  E  =  F G H I  >=  2100
and        388                >=  A B C
Therefore 388  *  A B C  *  E  >=  2100  *  A B C
So E > 5.4 that is E is 6 or 8.


326 328 346 348 * 26 = 8476 8528 8996 9048
326 328 346 348 * 28 = 9128 9184 9688 9744
326 328 346 348 *  8 = 2608 2624 2768 2784

  348 * 28 = 9744
  348 *  8 = 2784

  3 4 8
    2 8
------- *
2 7 8 4
6 9 6
-------
9 7 4 4


References:

-  Ch. W. Trigg, Mathematical Quickies, 1985, Problem 233.
   with explained solution

-  Zweisteins Zahlenlogeleien, Insel it 1510, Problem 21.

-  Zweistein, Zeit Magazin Nr. 35, 1989-08-25