# Two new chess pieces -- spin-off of knight

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.

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

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).