Substitute each of the small letters in bold with a different decimal digit from 0 to 9, to satisfy the following system of simultaneous modular alphametic equations:

one ≡ 1 (mod two), and:
one ≡ 1 (mod six), and:
six ≡ 0 (mod two)

where, none of s, o and t is zero.

Note: Try to derive a non computer-assisted solution, although computer program/spreadsheet solutions are welcome.

PRINT

FOR o = 1 TO 9
IF used(o) = 0 THEN
 used(o) = 1
FOR n = 0 TO 9
IF used(n) = 0 THEN
 used(n) = 1
FOR e = 0 TO 9
IF used(e) = 0 THEN
 used(e) = 1
 one = o * 100 + n * 10 + e
FOR t = 1 TO 9
IF used(t) = 0 THEN
 used(t) = 1
FOR w = 0 TO 9
IF used(w) = 0 THEN
 used(w) = 1
 two = t * 100 + w * 10 + o
 IF one MOD two = 1 THEN
FOR s = 1 TO 9
IF used(s) = 0 THEN
 used(s) = 1
FOR i = 0 TO 9
IF used(i) = 0 THEN
 used(i) = 1
FOR x = 0 TO 9
IF used(x) = 0 THEN
 used(x) = 1

 six = s * 100 + i * 10 + x
 IF six MOD two = 0 AND one MOD six = 1 THEN
   PRINT one; two; six
 END IF

 used(x) = 0
END IF
NEXT
 used(i) = 0
END IF
NEXT
 used(s) = 0
END IF
NEXT
 END IF
 used(w) = 0
END IF
NEXT
 used(t) = 0
END IF
NEXT
 used(e) = 0
END IF
NEXT
 used(n) = 0
END IF
NEXT
 used(o) = 0
END IF
NEXT

produces

 865  108  432

indicating one, two and six in that order.

Posted by Charlie on 2008-11-30 21:44:26