# Latin Squares

### Task 44 ... Years 2 - 10

#### Summary

Using a 3 x 3 grid and three blocks of each of three colours, arrange the blocks so that each row and column has exactly one of each colour. Try a 4 x 4 grid. Try a 5 x 5 grid.

#### Materials

• 25 coloured blocks or tiles - 5 each of five colours

#### Content

• patterns in 2D space
• arrangements and order
• problem solving strategies
• recording and explaining mathematics

#### Iceberg

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

Square A

Square B

The card suggests finding only one latin square of each size, but of course the challenge is in seeking others. One way to be sure of finding a latin square is to use this process:

1. Set up the first row with one of each colour.
2. Repeat it in the next row, then take the right hand one, place it at the left hand end and shuffle the others to the right one space.
3. Continue to repeat Step 2 until you make a square.
This approach is shown by the 3 x 3 drawing on the card and by Square A, but is it the only way to find a solution? Suppose we keep black as the first but swap the other two, then use the process above. The result is Square B. Is it different? How will you decide? What will be the criteria for same?
• What happens if black is not the first block in the first row?
The search for all the solutions of the 3 x 3 grid is now a serious problem.

Another process which could produce a solution is:

1. Place the first block in the left cell of the first row.
2. Bring another block of the same colour in from the left of the second row and slide it along to its first 'legal' position.
3. Repeat with the same colour in the following row(s).
4. Return to the first row and place the second colour in its first 'legal' position.
5. Repeat Steps 2 & 3 with this colour, but if an action is impossible retrace your steps, undo the previous move and make a different choice.
6. Continue this process of advance and backtrack if necessary, until that colour is completed, then go through a similar process with each of the other colours.
Square C
In Square C, all the white squares have not yet been coloured. Just look at the black and grey.
• The black have been placed using Steps 1 - 3.
• The greys have been placed using Steps 4 & 5, but there is an impasse. There is no 'legal' place for grey in the third row.
• Therefore the grey in the second row must be tried in the right hand position.
Doing so leads to Square A.

But suppose the black in the second row was placed in its second 'legal' position. Then the blacks would have to be placed like this:

Square D
• Does this lead to a solution?
• Is it a solution already found using the cyclic method above?
When you eventually discover how many solutions there are for the 3 x 3 Latin Square:
• Can you work out how many ways there are of placing the blocks so they don't make a Latin Square?
• Are you ready to explore the possible permutations of the 4 x 4 or perhaps the 5 x 5?
• What happens if we introduce the extra rule that both major diagonals must show each colour only once? This is the basis of Task 112, Coloured Squares.
• Can we build a 3 x 3 x 3 Latin Cube?

#### Latin Squares & Design

Both these designs were made from a Latin Square in the bottom right hand corner of the design.
• Find the starting square.
• What are its three pieces?
• Describe how the design has been made from the Latin Square.

Square E

Square F
What happens if the three 'colours' become pathways based on the trisection of each side of the unit square? For example:

#### 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 you only need plenty of blocks, tiles or Unifix cubes and grids to match. The questions above suggest the direction of the investigation.

At this stage Latin Squares does not have a matching lesson on Maths300.

#### 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 Latin Squares task is an integral part of:

• MWA Pattern & Algebra Years 5 & 6
• MWA Space & Logic Years 9 & 10
This task is also included in the Task Centre Kit for Aboriginal Students.