Charlie has been designing square 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 square tablecloths of other odd by odd sizes with just $1$ line of symmetry.
Check you agree that a $7$ by $7$ tablecloth can have at most 28 colours. Can you find a way of working out the maximum number of different colours that can be used on an n by n tablecloth (where n is odd), following Monday's rule?


Tuesday's $5$ by $5$ tablecloth has rotational symmetry of order $4$, and no lines of symmetry. Design some square tablecloths of other odd by odd sizes with rotational symmetry of order $4$, and no lines of symmetry.
Check you agree that a $7$ by $7$ tablecloth can have at most 13 colours.
Can you find a way of working out the maximum number of different colours that can be used on an n by n tablecloth (where n is odd), following Tuesday's rule?


Wednesday's $5$ by $5$ tablecloth has $2$ lines of symmetry (horizontal and vertical), and rotational symmetry of order $2$. Design some square tablecloths of other odd by odd sizes with $2$ lines of symmetry, and rotational symmetry of order $2$.
Check you agree that a $7$ by $7$ tablecloth can have at most 16 colours.
Can you find a way of working out the maximum number of different colours that can be used on an n by n tablecloth (where n is odd), following Wednesday's rule?


Thursday's $5$ by $5$ tablecloth has $2$ (diagonal) lines of symmetry and rotational symmetry of order $2$.
Design some square tablecloths of other odd by odd sizes with $2$ (diagonal) lines of symmetry and rotational symmetry of order $2$.
Check you agree that a $7$ by $7$ tablecloth can have at most 16 colours.
Can you find a way of working out the maximum number of different colours that can be used on an n by n tablecloth (where n is odd), following Thursday's rule?


Friday's $5$ by $5$ tablecloth has $4$ lines of symmetry and rotational symmetry of order $4$.
Design some square tablecloths of other odd by odd sizes with $4$ lines of symmetry and rotational symmetry of order $4$.
Check you agree that a $7$ by $7$ tablecloth can have at most 10 colours.
Can you find a way of working out the maximum number of different colours that can be used on an n by n tablecloth (where n is odd), following Friday's rule?

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.