Knight’s Tour Using Larger Multiples

A Knight’s Tour is a type of Hamiltonian path problem in graph theory. In the mathematical field of graph theory the Hamiltonian path problem are problems of determining whether a path in an undirected or directed graph that visits each vertex exactly once. 

In a Knight’s tour, a knight is placed on any square ("cell") on a rectangular grid and is moved until all cells are covered; a knight cannot visit the same square twice. On an 8x8 board, number each move from 1-64. You can start from any one of the 64 squares on the chess board. You do not need to know how to play chess; you just need to know how a knight moves. The knight moves in an "L" pattern; two squares horizontally or vertically; then one square to the right or left; knights may jump over other moves.

There are many strategies but one is to stay along the perimeter or outside of the board for as long as possbile and trying to reach each corner or vertex. If you are stuck, you can erase a few moves and see if there is an alternative route that will maximize your score for that tour. There are some very sophisticated graphing theory strategies that I will share with the children next class. 

An open tour or curcuit is reached when you reach the final cell but it is NOT a knight's move from the first cell. A more difficult closed curcuit is when you can reach the first cell from the 64th with a knight's move. Also, using a geometric pattern of connecting the center of each cell with a straight line can be quite beautiful. See the attached pdf.

The earliest recorded example of a Knight's Tour on the ordinary 8 x 8 board is described in an arabic manuscript with the title " Nuzhat al-arbab al-aqulfi'sh-shatranj al-manqul" (The delight of the intelligent, a description of chess) and came from al-Adli ar-Rumi, a professional chess player who lived in Bagdad around 840 A.D.  The pattern of a Knight's Tour has been presented in verse form in the highly stylized Sanskrit poem Kashmiri poet Rudrata.

WHAT IS NEW THIS YEAR? Every year I introduce a new concept in this Mathlete favorite activity. Instead of recording 1-64 you first choose a multiple such as 2. In the first cell you record a 2, then 4, 6, 8 ... 128=64x2. If you choose 5 as a multiple, it will start with 5 and end with 320=64x5. If you choose 5 as a multiple and get stuck at 100, that means you have made 20 moves (100/5=20).

I challenged older Mathletes to try fun multiples such as 7,13, 15, 17, 19, etc. (be careful not to choose too large a multiple so the 64th move will fit in your square. If you fill all of the squares, your last number should be 64 x your multiple. When the children check in with me to see how they are doing, I show them that you can take the last number and divide it by their multiple; if the quotient (answer to a division problem) is a whole number, they have not made any mistakes in their multiple counting. If the quotient has a remainder, they are encouraged to trace their steps to find the mistake. It is tedious work but extremely rewarding to find their mistake.

The attached pdf also has an example of the first magic knight's tour, found by William Beverley, published in 1848. It is shown in both arithmetical and
geometric forms. I encouraged the older Mathletes to follow the magic knight's tour with multiples of 2 or 3 or 4, etc.  Then they should up any row or column (not the diagonals) to make sure the totals are equal. These are the attributes of a magic square.

WARNING: This activity is addictive; you should try it as well. There have been reports of children doing Knight’s Tours well into the night in order to solve this ancient puzzle. Since I let out a scream of delight when solving my first tour, I gave them “permission” to do the same when they solve their first tour. I hope you will support this silly gesture.