Here is how Mathematica can be used to solve New Scientist Enigma number 1459 in the 8 September 2007 issue.
This is a straightforward digit manipulation problem which can easily and quickly be solved by a brute force search. In this problem there is no point in trying to do any clever programming.
Initialise the counter.
Step the counter until the condition on its digits is satisfied.
While[Not[MemberQ[Map[FromDigits, Permutations[IntegerDigits[n]]], 12n/11]], n++]
Display the counter value that satisfies the condition.