Date modified: Mon 2023-05-15 . 09:28 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Switch the knights. Warm up. A usual knight moves in a 2-by-1 $L$ shape. Consider two knights of one color (say black)) and two knights of another color (say white) are placed in a $3\times 3$ chess board below. Can you move them so the positions of the black knights and the white knights are switched? Note, knights shouldn't move into a square that is occupied. You don't have to move the pieces in alternating colors like in real chess, but try if possible. ![[---images/Switch the knights 2023-05-04 10.15.39.excalidraw.svg]] --- What about the following configuration, can you switch the positions of all the black knights with white knights? Can you move the knights in alternating colors like in real chess? ![[---images/Switch the knights 2023-05-04 10.12.36.excalidraw.svg]] Now, if you label the black knights on the top row $1,2,3$, can you switch the colors of the knights, so the black knights on the bottom, AND to be in any permutation that you desire? Which ones are possible? ![[---images/Switch the knights 2023-05-04 12.41.14.excalidraw.svg]] --- How about the following $4\times 3$ configuration, switch the knights. Can you move the knights in alternating colors like in real chess? ![[---images/Switch the knights 2023-05-04 10.08.06.excalidraw.svg]] --- Challenge. Now label the three black knights on the top row $1,2,3$. Can you switch the knights, and have the black knights on the bottom row but arranged in any order that you like? Which of the 6 permutations ($123,132,213,231,312,321$) are possible? --- Challenge. Find the minimum number of moves required to switch the knights in above $4\times 3$ configuration. ///hints to the $3\times 3$ case./// Sometimes chess problems are just graph theory problems...! ![[---images/Switch the knights 2023-05-04 12.52.07.excalidraw.svg]] /// ///hint to the $4\times 3$ case./// ![[---images/Switch the knights 2023-05-04 10.05.36.excalidraw.svg]] /// #puzzle #chess #graph-theory