# Math camp qualifying quiz 2014 Problem 1

#### MariaAgnesi

Hi I am hoping to participate in a math camp for gifted students next summer. In order to go there I have to pass a qualifying test that will be released soon. In the meantime I have the test from last year to practice with and I am really struggling with it. Any help would be greatly appreciated.

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.)

#### Math Message Board tutor

Hi I am hoping to participate in a math camp for gifted students next summer. In order to go there I have to pass a qualifying test
that will be released soon. In the meantime I have the test from last year to practice with and I am really struggling with it.
Any help would be greatly appreciated.

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.)
(Repeat Message)

If you're really gifted, then you need to demonstrate it here. Show some of your attempts here.
If you can't even show some of what you know about what you know here, then you're
probably not ready to pass that qualifying for that math camp. And you shouldn't go to the math camp.