# Chess Queens

### Task 150 ... Years 4 - 8

#### Summary

First students need to know how a chess queen moves. Then the challenge is easy to state. Place 8 queens on the board so that every square 'guarded' by at least one queen. The card shows that this challenge is so easy, that it is hardly worth considering. But what happens if we change the number of queens to 7 or 6 or 5? Can all squares still be guarded? And what happens if we change the size of the board?

#### Materials

• 8 markers to represent queens
• playing board

#### Content

• spatial thinking/visualisation
• logical reasoning and problem solving strategies
• generalisation
• algebraic patterns

#### Iceberg

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

There are several unique solutions for each challenge on the card and several more that are rotations or reflections of other solutions. Example solutions for each 7, 6 and 5 queens are:

 Seven Queens ... Six Queens ... Five Queens ...

The final challenge on the card remains open. We are waiting for one of your students to find a solution, or prove that there isn't one. We would be very happy to publish and credit their work.

#### Extensions

The card suggests that this problem can be explored by asking What happens if we change...?. The variables in the problem are the number of queens and the size of the board. The challenge might also be rephrased. Instead of asking for every square to be guarded, the more general question would be:
• What is the maximum number of squares that can be guarded?
If it is every square, as in the solutions above, the maximum is 64 (for an 8 x 8 board).

Altering these variables produces new investigation such as:

• If we are using an 8 x 8 board what is the maximum number of squares that can be guarded by 1, 2, 3, 4, 5 queens? (The answer to the last of these being 64).
• If we are using only 1 queen, what is the maximum number of squares that can be guarded on a 1x1, 2x2, 3x3, 4x4, 5x5, 6x6, 7x7, 8x8 board?
The results of this second question produce a pattern that can be represented by an equation. However, they also show that three pieces of data is not necessarily enough to predict a pattern.

#### 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.

To convert this task to a whole class investigation designed to model the work of a mathematician, you will need 2cm wooden cubes and grids to match their dimensions. These are easy to produce as tables in a word processor. If you don't have cubes, but you do have Poly Plug these Chess Queens Boards are provided for you. You will only need to print the 8x8, 7x7 and 6x6 boards as anything smaller can be represented on the red board. However, the boards will also work with 2cm counters hence all sizes to 2x2 are included in this PDF file.

The challenge can be introduced by analogy with the well known game Hide & Seek. Using a demonstration model on a central table, first check that students know how a chess queen moves.

So a queen is pretty powerful because it can stay in one place and guard many squares. Today we are going to explore the idea of hiding from the queen just like you hide when playing Hide & Seek.
Ask two students to help you demonstrate on an 8 x 8 board. Provide one with several blue plugs and the other with one yellow plug. The first student places 1 queen. The second looks for a place to hide from the queen and shows it by placing the yellow plug.
Okay, that was pretty easy. But can you find a place to hide if there are two queens on the board?
The hider will soon be able to feel safely hidden. Encourage the queens to be moved around to see if the first way of hiding was just a fluke.
Now you have the idea, the first challenge for the day is this. If there are five queens on the board, is it possible to place them so there is nowhere to hide? In other words, place them so that all spaces are guarded by at least one queen.
When one solution to this challenge is found, encourage the hunt for more before tackling the extensions above. Perhaps another copy of the boards can be used to record any solutions. The records can become a class display rather than being kept in student journals.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 129, Chess Queens, which includes software to assist the exploration of different size boards with various numbers of queens.

#### 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.

The Chess Queens task is an integral part of:

• MWA Space & Logic Years 9 & 10

The Chess Queens lesson is an integral part of:

• MWA Space & Logic Years 9 & 10
This task is also included in the Secondary Library Kit. Solutions for tasks in the latter kit can be found here.