### Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares?

### DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

### Sixational

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

# Attractive Tablecloths

### Why do this problem?

Rather than deducing an $n^{th}$ term rule by pattern-spotting, this problem encourages students to look at the structure of several symmetric patterns and to explain how to generate rules for finding the number of colours required.

### Possible approach

These printable resources may be useful: Attractive Tablecloths,
Attractive Tablecloths,
Templates.

Display these $5$ by $5$ tablecloth designs showing each of the five symmetry rules:

"This tablecloth is symmetric. Can you describe the types of symmetry it has?"

Once students have established what the five symmetry rules are:

"For each symmetry rule, I want you to find out the maximum number of colours you can use to colour in a $5$ by $5$, a $7$ by $7$ and a $9$ by $9$ tablecloth."

"In a while, I'll be asking you to work out the number of colours needed for a much larger tablecloth, one that's too big to draw. So you will need to record your results and think about how your work on these smaller cases could be generalised."

Hand out these templates and invite students to work in pairs using colours or numbering to create each design. As they finish each one, the designs could be pinned up together according to each symmetry rule, and students could look at each other's work to check there is agreement on the maximum number of colours needed.

"Imagine I wanted to create a $99$ by $99$ tablecloth. How many colours would I need for each type of symmetry? You need to develop a convincing argument to explain how to work out the number of colours needed. You might also like to consider how many colours would be needed for an $n$ by $n$ tablecloth."

Allow students time to work in pairs on this challenge. Then combine pairs and ask each group of four to refine their convincing argument for one or two of the designs. Finally, invite groups of four to present their convincing argument to the rest of the class, who could act as critical friends and suggest improvements to their explanations.

### Possible extension

Invite students to consider $n$ by $n$ tablecloths where $n$ is even.

A very challenging extension is to come up with a combined rule that works for both odd and even sized tablecloths, for each type of symmetry.

### Possible support

Seven Squares and Steel Cables both offer students the chance to deduce algebraic rules by looking at the structure of different patterns.