Knight’s Tour: Advanced and Guaranteed Tour in Four Quadrants

The children showed up to class with an extreme level of enthusiasm. Many children solved Knight’s Tours (1-64), some solved very rare Closed Tours (64th move to 1st is a knight’s move). Several  groups reported that they introduced Knight’s Tours to their school class and many of their teachers made hundreds of copies and distributed to other classes. These Mathletes were very proud.

This week, I wanted to continue this exploration but bring it to another level. I asked the children to solve a Knight’s Tour on a 3x3 board from 1-9 (I requested that they hold their verbal reaction to allow all of the children to discover). Some children could see immediately that it is impossible to ever reach the middle square but all were able to reach from 1 to 8 and then complete a geometric tour going back to 1 which formed a beautiful 8 pointed star. Then I had them split an 8x8 board into four quadrants (each a 4x4 board). They were asked to solve the 4x4 and many reached 15 but no-one could reach 16. I reported that I tried this challenge over 100 times using every possible permutation (combination) without success. This doesn’t mean it is not possible but it is unlikely. Then I challenged the to try a 5x5 knight’s tour. I simply gave them the task of starting in a corner or vertex and reach 25 in the center by following the widest perimeter as possible and continuing in the same direction. Almost all of them were able to solve this puzzle. The attached pdf. has 6x6, 7x7, and 8x8 boards. It also has a 16x24 board with an attached solution that is a closed tour reaching 384 in a beautiful pattern.

Now for the children above grade level 3, I introduced an algorithm to solve a knight’s tour on demand; and often a closed tour. This was very challenging but with a lot of patience is doable (see pages 10 and 11 on pdf). First split up the board into four 4x4 quadrants. Create dots in each quadrant that resemble either a non-square rhombus (most children call this a diamond; it is not a diamond, it is a quadrilateral with four equal sides) with a forward slope or a negative slope, or a square with a positive or negative slope. The positive slope is tilted south west to north east and the negative slope is tilted south east to north west. After you create the 16 dots (four for each figure), start to connect them by completing each quadrant (remember not to close the rhombus or square), and then moving as a knight would to the next quadrant. Once you complete the 16 moves, create the next shape (if you chose a non-square rhombus to start, next create a square as long as you can get to the next quadrant with a knight’s move. Create the 16 dots of the same shaped square and begin connecting until you fill all 16 dots, then continue with the other sloped non-square rhombus, and finally the other sloped square. Be careful to go the correct direction when starting in a quadrant so as to not get stuck too far away from the next quadrant. If you plan well, you should also be able to do a closed tour.

Enjoy as many of these challenges as possible and spread the Knight’s Tour word.