I grew up thinking that Ganesha broke off his own tusk to take dictation from Veda Vyasa. The story goes that Vyasa wanted to dictate an epic poem, the Mahabharata. He needed someone to write it down as it flowed from his mind. Nobody capable was found to exist on Earth, so he approached Ganesha or Vighna Vinayaka, the remover of obstacles, and requested him to help out. "Take dictation?" asked Ganesha. "If you please," said Vyasa. "Very well, I shall," said Ganesha, but added the proviso, "You may not stop at any point or I shall stop writing and go away."
We were taught that the gods love to twist the odds for men just to see how they will react, since all creation is their divine play or lila. Since Vyasa was clever - he composed a long and complicated epic like the Mahabharata - he made a counter-condition that Ganesha had to understand every word before he wrote it down. Ganesha was pleased to agree, secretly delighted. He broke off his own tusk to write with, as proof of goodwill, and they began the task.
Every now and then, Vyasa would compose a number of verses in extremely layered and dense language. Ganesha would have to pause to think them through before putting them down on the palm-leaf pages. This allowed Vyasa to draw breath and compose more verses in his head to stay ahead.
Vyasa's response to Ganesha's proviso was pointed out as an example of an organic solution to a problem; Vyasa had come up with this pragmatic strategy because he was goal-oriented and intent on fulfilling his mission. This legend was a popular teaching story to inspire focus and concentration; and its iconic reminder in daily life was the broken tusk on all Ganesha idols.
Another myth about the tusk that I saw danced out in Thailand in December 2012 occurs in the Brahmanda Purana. It was taught to the young to instil absolute obedience. Parashurama or Rama of the Axe was Lord Vishnu's sixth avatar, believed to have never really gone away but to still be meditating on Earth in a secret cave in the mountains.
Lord Shiva had given Parashurama the axe to help him complete his mission. When Parashurama's task was over, he made his way to Mount Kailash where Shiva lived, to return the axe and thank him. But at the mystic mountain, he found his path blocked by Ganesha. Lord Shiva had ordered his son to guard Mount Kailash against all visitors, since he was about to go into a long, deep trance and did not want to be disturbed.
Parashurama, a Shiva-bhakt, was so furious at being blocked by Ganesha that he threw the axe at him. Ganesha could have easily deflected the axe. But since it belonged to his father, he did not stop it, out of respect, and let it break his tusk. It is intriguing to think how these ancient stories animated so many minds across space and time, and continue to do so.