MAIN CONCEPTS
Apply the function three times to k making sure you consider the cases when n is odd or even.
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First and foremost do not get intimidated by the ‘function of a function of a function’. This question is easier than it looks. All it requires is a careful implementation of the function . What you have to pay close attention to is the function is defined differently based on whether n is odd or even.

Since  is an odd integer, we have , which is an even integer. So

But now there are two possibilities to consider. We have no way of knowing if this last value is even or odd. We consider the cases separately.
                             
If  is even, then

If  is odd, then

 

So which of these values is correct. We will simply work recursively putting these values into the original function  and see whether it leads to 27.

For ,

which is correct.

For

which is incorrect.

So the correct value is , whose sum of the digits is 6. This questions the irregularities that occur when applying the function continuously giving us a result which does not necessarily satisfy the initial condition.