Is it possible to free Todd of his curse?
It seems it is just a story mechanism. There is no where near enough unique hairstyles to fulfil this requirement yourself.
Even if you included colour variants, you could only achieve 12 styles * 6 colours, which is 72 possible combinations.
I guess this is why it explicitly states "hairstyles", so that the curious user doesn't waste any time in trying all the possible combinations for overall makeovers. Which would be well over 100,000 combinations (I don't know how many eye colours there are to be more specific)