Knight Protectors

Task 153 ... Years 4 - 10

Summary

The problem is easy to state and easy to start, but requires a combination of spatial ability, recording and reasoning to solve. Place twelve knights on a board so that every square is either protected or occupied. The task is a partner to Knight Swap in that it is based on the movements of a knight in chess and its solution involves rotation. In fact, it is rotationally symmetric.
 

Materials

  • 12 'knights' and a board

Content

  • logic & reasoning
  • spatial perception
  • concept of rotation
  • recording position on a grid
Knight Protectors

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
   

The pen and cloth are provided so students can 'cross out' the protected squares as they place each knight. As they record the image of the 'ring' of protected squares connected to each knight is reinforced and the part 'rings' associated with knights near the edge of the board becomes apparent.

It is common for students to start in the middle and develop something like this where black is a knight:

    
The blue shows that the 'ring' pattern of protected squares continues with the second night and 'moves' one square to the right to match the placement of this second knight.

Continuing with a third night in centre adds the green squares and two more that are already protected by a previous knight. Which ones? Continuing with the fourth knight adds the orange ones to the collection of protected squares.
    

The protected squares are starting look encouragingly symmetric - and so are the unprotected ones. There are 8 knights left to place and symmetry suggests if this approach is going to work, two will be needed to protect each quarter. At this point students might divide the problem into smaller parts and consider just the top left quarter.

The problem section is likely to be the corner square, so let's get that used first. One way is to put a knight in that square. Now it is occupied and it protects ... just two other squares that are already protected (the yellow ones). The one knight left can't possibly stretch its resources to protect the remaining vacant squares in this quarter.

Conversely though, to protect the corner square, one of the knights must be in one of the two yellow squares. Now students can make an organised search using 'if-then reasoning' to determine if it is possible to place the two knights in this quarter.

It isn't!

However, all is not lost, because the symmetry suggests another approach. If we could start again and solve the problem by protecting or occupying every square in one quarter, then that pattern could be replicated in each of the other quarters. It would also imply the need for three knights in each quarter.

Again there is exploration involved, and again the difficulty of using the corner square must be considered, but eventually this arrangement proves promising.


(Colours have been added in the same sequence as above - yellow, blue, green.)
It's true that we have not protected or occupied every square in the top left quarter, but:
What happens if we now move to the top right quarter and start by placing a knight that will protect the most 'inland' of the unprotected squares in the top left quarter?
Of course the top right corner square has to be considered too, but continuing this type of reasoning should lead to:
which solves the problem and looks rather beautiful at the same time. (There are variations on this solution, but are they different?)

Extensions

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
   

A good way to capture interest in this problem is to introduce it with a large 8 x 8 grid that can be placed on a central table or floor space. A grid of squares with sides 10 - 12cm works well. The art room will probably be able to supply large size paper to start you off. Given the reasoning above, it is also useful to make the big grid from four 4 x 4 grids and masking tape them together on the back. The big grid can then be folded into sections if needed. You will also need 12 identical objects (milk cartons perhaps?) to represent the knights and other objects, such as counters, to represent the protected squares. If you have access to 10cm squares of coloured paper, called Kindergarten Squares in some places, they work really well to show the protected squares - provided there isn't a fan or air-conditioner blowing a gale in the classroom!

Gather the students around the board and check first that all are clear about how a knight moves. Place one knight on the board and ask various students to place coloured paper to show the squares it protects. This will illustrate the symmetry indicated by the diagram on the task card above. Add another knight and another set of coloured squares. Discuss the connection between the two sets of protected squares.

It's like the knight has a skirt that touches in a sort of ring around it. The second knight shows that the touching pattern stays the same if the first knight moved to this new place.
You might also explore one more knight to show that there are some positions where the skirt touches outside the 8 x 8 grid.

Now explain the main challenge and invite students to return to their tables to work in pairs using graph paper and objects such as 1cm cubes. This Knight Protectors Paper provides each pair with four chess boards to experiment with using these cubes. As students explore, look for opportunities to share insights - probably by returning to the central grid - and let students lead the way to the solution above.

Highlight the rotational symmetry of the solution and invite students to record the key elements of this investigation in their journals. This could be a personal 'diary style' entry, or the investigation could be used to model the process a mathematician has to go through to publish their findings for others. This will require scaffolding and supporting the writing process using techniques such as those in one of the two lessons on Learning to Write a Maths Report.

At this stage, Knight Protectors does not have a matching lesson on Maths300, although it could be used in a unit with Lesson 82, Knight's Tour, which involves moving a single knight around the chess board so that it visits every square exactly once.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
   

Knight Protectors is not in any MWA kit. However it can be used to enrich the Space & Logic kits at Years 5/6 and 7/8.

Green Line
Follow this link to Task Centre Home page.