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# Attractive Tablecloths

##### Age 14 to 16Challenge Level

Attractive Tablecloths printable worksheet - tablecloths
Attractive Tablecloths printable worksheet - templates

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.