The Four Knight's Move Puzzle
Courtesy, The Lilly Library, Indiana University, Bloomington, Indiana.
Starting with two black and two red Knights as positioned in the image below, show how to exchange the positions of the black and red Knights in seven moves using legal Knight's moves.
Knight's Move Solution by Dan Thomasson, March 5, 2002
The trick to this puzzle is knowing that the term 'move' does not constrain the Knight to one jump, elsewise there would actually be 16 separate moves. Also, alternating red and black Knight's moves does not appear to be required thus breaking the basic rules of chess. This puzzle was published in 1512 by Paolo Guarini di Forli (best known simply as Guarini) and normally labeled: "The Puzzle No Mortal Can Solve."
If you look at the actual number of jumps by each Knight, they go in the following sequence: 1 - 2 - 3 - 4 - 3 - 2 - 1 (16 jumps, 7 moves). If you take the square root of 1234321, you end up with 1111. I'll let your imagination figure out various ways to use the Knight's Move in math (octal/binary) and cryptography. If you connect the lines for the odd moves (1, 3, 5, 7) or the even moves (2, 4, 6) only, they both make a complete 8-pointed star.
For the purist at heart that wants to solve this puzzle in the fewest legal Knight's Moves and alternating between color of Knights, the solution requires 16 moves.
If you look at the actual number of jumps by each Knight, they go in the following sequence: 1 - 2 - 3 - 4 - 3 - 2 - 1 (16 jumps, 7 moves). If you take the square root of 1234321, you end up with 1111. I'll let your imagination figure out various ways to use the Knight's Move in math (octal/binary) and cryptography. If you connect the lines for the odd moves (1, 3, 5, 7) or the even moves (2, 4, 6) only, they both make a complete 8-pointed star.
For the purist at heart that wants to solve this puzzle in the fewest legal Knight's Moves and alternating between color of Knights, the solution requires 16 moves.
A Four Knight's Solution
There are several different solutions to the Four Knight's Move puzzle. But what is the easiest way to derive at a solution without using trial and error? Graph Theory. What is Graph Theory? As per Wikipedia:
- "In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics."
Knight's Star Moves
Unfolded Knight's Star Moves
With White Knights on 'a1' and 'c1' and Black Knights on 'a3' and 'c3', simply rotate around the star clockwise, or counterclockwise, beginning with a White Knight's Move, then a Black Knight's Move. Continue the moves around the star by alternating between White and Black Knight's moves. Write down the move sequence of each Knight to derive at the correct solution. Apply the moves on a real chessboard, or an online chessboard, to see how easy it is to solve the puzzle without using trial and error.
Unfolded Knight's Tour Moves keeping original lines and making a 4th Dimension Hypercube.
A correct solution is provided in the above animation starting with A1 (White Knight) moving to C2. Then the Black Knight moves from C3 to A2 clockwise around the board. It is more difficult to see the moves in this image, but they are the same moves as shown in the graphic called 'A Four Knight's Solution.'
Here is a list of all the moves in this image that provides a solution for the Four Knight's Move puzzle. The White Knight's Move is listed first on the left with the Black Knight's Move listed to the right. This is similar to the way chess notation is done during chess tournaments. For clarification, I am showing the beginning square and landing square of each move. 'N' represents the Knight in English chess notation and not 'K' which is reserved for the King.
Here is a list of all the moves in this image that provides a solution for the Four Knight's Move puzzle. The White Knight's Move is listed first on the left with the Black Knight's Move listed to the right. This is similar to the way chess notation is done during chess tournaments. For clarification, I am showing the beginning square and landing square of each move. 'N' represents the Knight in English chess notation and not 'K' which is reserved for the King.
- Na1-c2 - Nc3-a2
- Nc1-b3 - Na3-b1
- Nc2-a3 - Nb1-c3
- Nb3-a1 - Na2-c1
- Na3-b1 - Nc1-b3
- Na1-c2 - Nc3-a2
- Nb1-c3 - Nb3-a1
- Nc2-a3 - Na2-c1
Notice that in chess notation, there are only 8 moves listed above. Hmm, so if we are thinking of chess moves as if we were playing in a chess tournament, the correct solution would be 8 moves instead of 16. A move is not complete in chess until both a White piece and a Black piece each makes a move.
If you really want to get tricky about how many moves does it take to switch the position of the White and Black Knights on a 3x3 board, simply turn the board around thus making zero moves for the solution.
If you really want to get tricky about how many moves does it take to switch the position of the White and Black Knights on a 3x3 board, simply turn the board around thus making zero moves for the solution.