Knight’s Tour: Ancient Math Challenge

Knight’s tours were invented around the time of the introduction of chess. The earliest recorded knight’s tour on a 8 x 8 board was in a book published by Iraqi mathematician al-Adli ar-Rumi in 840 A.D. Many of the greatest mathematicians have explored knight’s tours like Euler, Taylor, de Moivre, and LaGrange.

You do not need to know how to play chess to do a knight’s tour. You simply need to know how a knight moves on a chess board. The knight moves in an L shape of 2 squares horizontally or vertically and then one square to the right or left. Knights may also jump over other pieces or moves.

We studied this challenge by first mastering the knight’s move by using chess sets and several knights. I recommend that students count “one, two” as they move the piece horizontally or vertically and then “over one” as they move to the right or left. Think of starting the piece by moving east, west, north and south first two squares and then making a turn to the right or left one square.

A knight’s tour is only solved if you fill all of the squares on the board without repeating a square. Place a number 1 anywhere on a grid and start moving the knight by numbering consecutively 2, 3, 4, and so on. The best strategy in the beginning  is to start in a corner and move along the perimeter (the outside of the board). We first looked at a knight’s tour on a 3 x 3 board and the students quickly concluded that if you start on the perimeter, you can only get to 8 squares and never complete the tour in the middle square. I had the students struggle with 4 x 4 boards only to find that the best they could do was up to 15 of the 16 squares.

The smallest square knight’s tour is on a 5 x 5 board. The children were able to solve this challenge by placing the number one in a corner and placing the number 25 in the center; if they move along the perimeter in the same direction, this is a simple tour. In the attachment called Knights Tours 2014.pdf, I provided the students with 3x3, 4x4, 5x5, 6x6, 7x7, and 8x8 boards as well as solutions to all 5x5 and up. 

I would like them to focus on 8x8 knight’s tours. If the children reach 64, and this square is a knight’s move away from square 1, it is called a closed knight’s tour or closed circuit. If the 64th square is not a knight’s move away from square number 1, it is an open tour. There are  over 26 trillion known closed knight’s tours on a 8x8 board but the number of possible open knight’s tours is still unknown.

The children and you will be addicted to this challenge so I provided them with several sheets each with ten 8x8 boards. I have also attached two documents with directions and four 8x8 boards and six 8x8 boards.

Since I screamed out of joy the first time I solved a knight’s tour, waking up my whole family, I jokingly gave the children permission to do the same. Hopefully, they will solve this puzzle during normal hours.