Why does Lord Ganesha have a half-broken tusk?

When he came up with the idea to Mahabharatha, Sage Ved Vyasa looked around for someone who would write down the words even as he recited them. He approached Lord Ganesha and asked him if he would do so. Lord Ganesha readily agreed.

Ved Vyasa said "I will narrate the story and you must take it down as fast as I say it and should not interrupt or stop anywhere in between" and Lord Ganesha too said that "I will do so and if you halt or hesitate, I will stop writing and your epic will never be written". Ved Vyasa agreed with his final condition "So I agree but you must also agree to fully comprehend the meaning of the poems as you write and not just blindly write them as I say".

The elephant headed God gave his consent and they started writing the epic Mahabharata. To write this Lord Ganesha broke a bit off his tusk and used it as a pen. As they continued with the writing, Ved Vyasa found that Ganesha was very fast and he didn't even have any time to breathe in air! His face slowly turned blue and he wondered how will he go on without pausing to draw his breath also. Then he came up with the solution and narrated a difficult stanza. Ganesha, who was writing at a furious pace, stopped for a second to get the meaning of the stanza and Ved Vyasa thankfully used the split second gap to gulp deep lungful of air. But Ganesha carried on immediately and they went on. And so, whenever Ved Vyasa needed a break, he would tell a difficult phrase and used the time Ganesha took to comprehend the meaning. Thus the original Mahabharata contains many difficult stanzas placed at intervals throughout the length of the epic.