Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Nim

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

Loopy

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

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."

If a computer room is available, students could work in pairs using the interactivities to design tablecloths according to each rule. The interactivities can be used to check that the maximum number of colours has been reached each time.

Alternatively, if students are working in a classroom without computers, 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

The last interactivity invites 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.