Suppose there is a species of aliens called cyborgs. There is an infinite chessboard in their homeland. There is 1 cyborg on every square. If cyborgs can jump infinitely far, or jump and land on their same square, is it possible to get 2 cyborgs on the center of every square after 1 jump (a jump is when every cyborg on the chessboard jumps).
another question: What if the cyborg can only jump a finite amount of squares, say 10, or 100 squares?
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If each glob can jump at most distance $r$, only those that start out within distance $R+r$ of the origin can end up within distance $R$ of the origin. But as $R \to \infty$, the number of globs that start out within distance $R+r$ of the origin is $\pi (R + r)^2 + O(R) = \pi R^2 + O(R)$. If after the jump there were two per square, the number within distance $R$ of the origin would have to be $2 \pi R^2 + O(R)$. So unless the globs reproduce, this is impossible.
Label the rows on half of the board $0,1,2,\ldots$. Let the cyborg in row $i$, column $j$ jump to row $\lfloor i/2\rfloor$, column $j$. Do something similar for the other half of the board.