A clever problem with a clever little answer.
Imagine all of the 35 seats are coloured like a checkers board. So they alternate colours white - black - white... along the rows and columns. I started with a black chair in the corner.
If you count up the number of each colour you will see that there are now 18 black chairs and only 17 white chairs.
Now, when every person moves 1 seat to the side they must swap the colour of their seat, as they are alternating. As you can see this is impossible, as someone from a black chair will not have a white chair to sit on.
So, the answer is - impossible!