Knights Tours (December 20-23)

All of the Mathletes learned about or rediscovered the mystery and challenge of Knights Tours.  This is an ancient puzzle that challenges us to place a number anywhere on an 8x8 grid and move as a Knight on a chess board (2-1 L shape) until you reach 64. As only a Knight can do, it may jump over other numbers but may not repeat a square. If you are stuck, use the method of recursion where we erase the last number on which we hit a wall until we can travel more moves in a different direction.

Please send me an email when you solve a Knights Tour of 64. Now for the real challenge: can your Knights Tour end on 64 one knights move away from the first square?  This is called a closed Knights Tour.  Use the attached excel spread sheet to work on your tours and share this with your family.

Once you solve a Knights Tour, next to your solution, use an empty grid to copy your tour using just line segments from the middle of each square to the next knights move in order to create a beautiful geometric design.

I also gave most of the classes a packet of 5x5, 9x9, 13x13, etc. grids (these are ordered 4k + 1).  These knights tours can be created by starting in a corner and moving along the perimeter (two rows around the edge makes up the perimeter in a knights tour) only until the last move above your corner starting point.  At this point you move to the third row from the edge which begins the next 4k+1 ordered square.  For example, when you start with a 9x9 square, your 57th move is the first move of your 5x5 tour. See the example on page 12 of the attached pdf. This pdf also has some amazing Knights Tours and closed tours created by famous mathematicians and two closed tours created by my son, Michael Kramer. Incidentally, I have not accomplished a closed Knights Tour yet; and I stress the word "yet."

AttachmentSize
knights_tour.xls34.5 KB
Knights_Tour_Many.pdf631.71 KB