### 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

##### Stage: 4 Challenge Level:

Charlie has been designing tablecloths for each weekday. He likes to use as many colours as he possibly can but insists that his tablecloths have some symmetry.

The $5$ by $5$ tablecloths below each satisfy a different symmetry rule.

 Monday's $5$ by $5$ tablecloth has just $1$ line of symmetry.    Use this interactivity to design tablecloths of other sizes with just $1$ line of symmetry.   Can you find a way of working out how many colours would be needed for an n by n tablecloth (where n is odd)? Tuesday's $5$ by $5$ tablecloth has rotational symmetry of order $4$, and no lines of symmetry.     Use this interactivity to design tablecloths of other sizes with rotational symmetry of order $4$, and no lines of symmetry.   Can you find a way of working out how many colours would be needed for an n by n tablecloth (where n is odd)? Wednesday's $5$ by $5$ tablecloth has $2$ lines of symmetry (horizontal and vertical), and rotational symmetry of order $2$.     Use this interactivity to design tablecloths of other sizes with $2$ lines of symmetry, and rotational symmetry of order $2$.   Can you find a way of working out how many colours would be needed for an n by n tablecloth (where n is odd)? Thursday's $5$ by $5$ tablecloth has $2$ (diagonal) lines of symmetry and rotational symmetry of order $2$.     Use this interactivity to design tablecloths of other sizes with $2$ (diagonal) lines of symmetry and rotational symmetry of order $2$.   Can you find a way of working out how many colours would be needed for an n by n tablecloth (where n is odd)? Friday's $5$ by $5$ tablecloth has $4$ lines of symmetry and rotational symmetry of order $4$.    Use this interactivity to design tablecloths of other sizes with $4$ lines of symmetry and rotational symmetry of order $4$.   Can you find a way of working out how many colours would be needed for an n by n tablecloth (where n is odd)?

EXTENSION
At weekends Charlie likes to use tablecloths with an even number of squares. Investigate the number of colours that are needed for different types of symmetric $n$ by $n$ tablecloths where $n$ is even. You may wish to investigate using this interactivity.