Knight’s Tour

The Knight’s Tour is a classic problem in chess and mathematics.

Work in pairs or small groups.  Difficulty Medium

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THE TASK

A knight starting at a1 visits all the squares on the board once and returns whence it started. Choose the first move as Na1-b3.

Allow the class to solve the problem using their own ideas without any further input. After trying for a while, they will realise that they need some way to identify the squares which have already been visited.

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Have a set of numbered counters handy as they are the most convenient way to track the knight moves.

Sort the numbered counters into the correct order. Place a counter on each square the knight visits. The path of the knight may be tracked by the sequence of the counters.

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GROUP WORKING

Let each group attempt to complete a tour. Suggest they divide up the tasks. For example, one person sorts out the counters, another is in charge of the knight, another places the counters as the knight leaves each square, with others giving support. Co-operation involves talking to each other, so expect chatter.

Invariably some groups will fail to complete the tour. There will be a few gaps on the board. Inform the groups that success is relative. Groups may compete with each other. The group which has the lowest number of empty squares is deemed the “winner”. Do not despair if you cannot complete a tour - you might have done better than the others.

Measure progress according to the number of visited squares. Any number above 50 is good; above 56 is very good; above 60 is excellent.

SYSTEMATIC SOLUTIONS

Encourage the groups to find a systematic solution method. Typical initial attempts involve rules such as “move around in a spiral” or “fill in one corner first”. Whilst groups are working on a solution, ask them to explain what method they used.

The “edge-hugging” method is straightforward but requires concentration and accuracy. It comprises a Main Rule with a Tie-Break.

Main Rule: Keep as close to the edge as possible;

Tie Break: If two moves are equally close to the edge, take the one closer to a corner.

Tell the groups to try this method. Clarify what the term “tie-break” means. Ask if they are aware of other tie-break methods e.g. in football goal-difference is often used if game points are equal.

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EXPLORATIONS

Before they get underway, ask the groups to suggest the first two moves.

Should the second move of the knight be to c1 or a5? Most groups choose 2.Nb3-a5. It seems they are naturally drawn to continue in the same direction as their first move.

Unfortunately, 2.Nb3-a5 is contrary to the Tie-Break rule. Both c1 and a5 at at the edge of the board, so the tie-break is engaged. c1 is two squares from a corner whereas a5 is three squares from a corner. The correct move is 2.Nb3-c1.

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