1. Imagine a chessboard that extends infinitely far in all directions. In the center of every square is a grasshopper.

(a) Show that it is possible for all the grasshoppers to jump simultaneously, so that after the jump there are exactly

two grasshoppers in the center of every square. (A grasshopper can jump arbitrarily far; it can also jump straight

up and land on the same spot it started on.)

(b) Now suppose the squares are 1 inch Ã— 1 inch, and a grasshopper can jump at most 100 inches. Is it still possible to

end up with two grasshoppers in the center of every square after one jump? If yes, show how it can be done. If no,

prove your answer. (Note: Saying that your strategy from part (a) wonâ€™t work is not a proof that it canâ€™t be done,

because there might be a different strategy that does work.)