1. Seven points on a circle are numbered 1 to 7 in the clockwise direction. A grasshopper jumps in the counterclockwise direction, from one point to another on the circle. If the grasshopper is on an odd-numbered point, it moves one point, and moves two points if it is on an evennumbered point. If the grasshopper begins at the point 7 , where will it be after 2011 jumps?
(ans. At the point 2 .
It will only land at the points $6,4,2,7$, landing at 7 after every fourth jump. $2011=3(\bmod 4)$, so it is at 2 by the 2011th jump.)