# Eight Queens

### Task 134 ... Years 4 - 10

#### Summary

A recreational puzzle from chess that opens into a considerable investigation involving every aspect of the work of a mathematician. The challenge is to place eight queens on the board so that none share the same row, column or diagonal with any other. Although there are many solutions to the task, finding the first can be quite a challenge. Students will have to give up 'guess and check' and try more refined reasoning to be able to solve the problem.

#### Materials

• Playing board
• 8 markers

#### Content

• spatial thinking (horizontal, vertical and diagonal)
• symmetry properties of reflections and rotations
• Working Mathematically process
• developing a range of problem solving strategies including:
• break the problem into smaller parts
• try every possible case

#### Iceberg

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

Apart from 'guess, check & improve', one approach is to break the problem into smaller parts by placing a queen on the top row first. Then a queen on the second row in the first 'safe' square; then the third row queen in a similar way; and so on. When a 'contradiction' arises, the problem has to be 'undone' to the last placement which had a choice. From that point the next choice is followed as far as possible - hopefully to completion of the problem, but if this doesn't happen, then retracing steps to the previous point of choice, begins again.

Finding solutions to this problem takes considerable patience and it is reasonable when using the task to be satisfied if the students find and record just one. However, ensure that they also record that there could be more. Then they will know that they could return to the task and look for another at any time - just as a mathematician would. Doing this also leaves the door open for the class as a whole to tackle the problem.

An additional feature is to ask each successful team to record their solution as part of a growing class display of solutions. This will show up that not all solutions are different. Many are reflections or rotations of each other. There are only 12 unique solutions. Here is one to start you off.

It could be recorded as 15863724. However, the two below would not be considered new solutions. The first is a reflection of this one about the right hand side and the second is a 90° clockwise rotation about the bottom right corner.

If you do count all the reflections and rotations as different, then there are 92 different solutions.

#### 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.

If an 8 x 8 grid exists, or can be created, outside or in the hall, it is really worth beginning the investigation by physically involving the students as the chess queens. Looking along horizontal, vertical and diagonal lines and seeing another student drives home the key element in the challenge. A class of 24 students can be separated into three teams of 8 and each is given five minutes in turn to find a solution. Build on any strategies that begin to show up and work towards the students considering the strategy of breaking the problem into smaller parts as described above. At the appropriate time move inside and issue grids and markers so that pairs can continue the search.

If it is necessary to justify why this lesson is being used, it's really quite easy.

We are learning to work like mathematicians and because this challenge doesn't require extra knowledge of numbers or algebra or such, we can focus on the process a mathematician uses to reason through any problem. This process is transferable to all content areas and has helped to create our understanding of number, algebra and those other things.
Or, perhaps more simply, this problem is exactly what mathematicians do.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 119, Eight Queens, which includes an excellent piece of companion software.

#### 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 Eight Queens task is an integral part of:

• MWA Space & Logic Years 5 & 6
• MWA Space & Logic Years 9 & 10

The Eight Queens lesson is an integral part of:

• MWA Space & Logic Years 9 & 10