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?

Pentagon

Find the vertices of a pentagon given the midpoints of its sides.

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.

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 determine 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 determine 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 determine 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 determine 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 determine 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.

Manipulating algebraic expressions/formulae. Mathematical reasoning & proof. Generalising. Reflections. Graph sketching. Interactivities. Making and proving conjectures. Symmetry. Rotations. Creating expressions/formulae.

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Attractive Tablecloths

Stage: 4 Challenge Level: