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