In one room of the science fair were six projects, and each project was entered by a pair of students. During the fair, some of the students visited other projects in the room and shook hands with some of the other project entrants. No students shook hands with there own partners. After the fair, the students discussed the projects they had visited. Hannah learned that no to of the other eleven students had shaken the same number of hands. How many hands did Hannah's parter shake?

please help me with this problem i keep trying but i keep messing up.

its due tomorrow and i cant find the solution