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An elf moves one square over and another 3 perpendicularly. An elephant moves two squares over and another two perpendicularly.
An elf is on square a1 of a regular chessboard and after a certain number of moves it return back to a1. Prove that it made an even number of moves.
An elephant is on square a1 of a regular chessboard and after a certain number of moves it return back to a1. Prove that it made an even number of moves.
I tried to solve the problem by attempting to show all the possible squares that the pieces could reach but that had too many cases and took forever. However, I do know that both pieces stay on the same color. I am not sure how to use this piece of information.

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    $\begingroup$ Looks like you need to recolor the squares that originally have the same color as a1. $\endgroup$ May 27, 2017 at 20:51
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    $\begingroup$ You could describe the elephant also as a bishop with a 2-square range. $\endgroup$
    – Kaj Hansen
    May 27, 2017 at 20:55

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For the elf, colour ranks 1,3,5,7 white and ranks 2,4,6,8 black. A move must change the colour of the square the elf is on, since it always moves an odd number of squares up or down.

For the elephant, it will never land on ranks 2,4,6,8, but you can colour ranks 1,3,5,7 alternating white and black.

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Mark the chess board using $2D$ coordinates, mark $A1$ by $(0,0)$ and proceed naturally. For the elf notice that after each move of the elf the $x$-coordinate and the $y$-coordinate both changes it's parity, initial and final coordinates are even(same parity), so number of steps must be even.

If the elephant moves then $+2$ or $-2$ is added to $x$-coordinate and $y$-coordinate of the previous position coordinate, so if the number of moves is odd, the number of $+2$ added to the $x$-coordinate can't be same as the number of $-2$ added to the $x$-coordinate, hence their sum can never equal zero(It must equal $0$ as we are reaching at $(0,0)$ finally).

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