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# Attractive Tablecloths

##### Age 14 to 16 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.    Design some tablecloths of other sizes with just $1$ line of symmetry. Check you agree that a $7$ by $7$ tablecloth needs 28 colours.   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.     Design some tablecloths of other sizes with rotational symmetry of order $4$, and no lines of symmetry.   Check you agree that a $7$ by $7$ tablecloth needs 13 colours.   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 $2$.     Design some tablecloths of other sizes with $2$ lines of symmetry, and rotational symmetry of order $2$.   Check you agree that a $7$ by $7$ tablecloth needs 16 colours.   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 $2$.     Design some tablecloths of other sizes with $2$ (diagonal) lines of symmetry and rotational symmetry of order $2$.   Check you agree that a $7$ by $7$ tablecloth needs 16 colours.   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 $4$.    Design some tablecloths of other sizes with $4$ lines of symmetry and rotational symmetry of order $4$.   Check you agree that a $7$ by $7$ tablecloth needs 10 colours.   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.