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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?

user138062
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  • Hint: There is a one-to-one correspondence between an infinite set and two disjoint copies of that set. – Robert Israel Mar 26 '14 at 03:25
  • The first question is a duplicate of http://math.stackexchange.com/questions/753354/prove-that-there-are-two-frogs-in-one-square – Barry Cipra Apr 17 '14 at 21:31
  • @BarryCipra, looking at the index numbers, the duplication would be the other way round wouldn't it? Not sure whether or not it matters. – David Apr 17 '14 at 22:33

2 Answers2

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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.

David
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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.

Robert Israel
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  • Where would you get the pi from? Sorry if this is obvious. – user138062 Mar 27 '14 at 03:17
  • @user: He's assuming the length of a jump is measured by Euclidean distance (you didn't specify), thus the set of squares less than a given distance from a given square resembles a disc. –  Apr 13 '14 at 04:00