Why does Shanmukha (Karthikeya) have six heads?
So that he can recite all the Shad Darshanas (six Hindu philosophies) at once.
Six faces - this has a significance. We say, 'The four directions', North, South, East and West. Brahma has four faces, which means that, he gets knowledge from all the four directions.
But Karthikeya is special, he has six faces. Sixth face is an indication of the sixth sense. It can also be interpreted as, he get knowledge from all the four directions and from above and below. In this way, you can interpret it in any manner.
Then there are six imperfections (vasanas) which trouble the human mind. Karthikeya is there to remove these six imperfections, i.e., Kama (lust), Krodha (anger), Lobha (greed), Moha (attachment), Ahankar (ego), and Matsarya (jealousy).
These six imperfections are removed by the six faces of Karthikeya. These are the six enemies, and these six enemies are eliminated by the six faces of Karthikeya.
There are six Darsanas:
1. Samkhya (strongly dualist theoretical exposition of consciousness and matter)
2. Nyaya (explores sources of knowledge)
3. Yoga (a school emphasising meditation, contemplation and liberation)
4. Vaisheshika (school of atomism)
5. Mimamsa (school of orthopraxy (emphasis on conduct))
6. Vedanta (knowledge in the Vedas)
Then there are six Vedangas (Limbs of the Veda):
1. Jyotisha (astronomy)
2. Chandas (meter)
3. Nirukta (etymology)
4. Vyakarana (grammar)
5. Shiksha (phonetics), and
6. Kalpa (ritual)
These are the six Vedangas and the six Upangas, and Karthikeya is a symbol of imparting this knowledge.