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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.
 
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.
 
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 $2$.
 
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 $2$.
 
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 $4$.
 
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.