Attractive Tablecloths

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
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Problem

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.

 

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Attractive Tablecloths

 



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?
 

 

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Attractive Tablecloths

 



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?
 



 

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Attractive Tablecloths

 



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?

 

 

Image
Attractive Tablecloths

 



 
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?

 



 

Image
Attractive Tablecloths

 

 


 
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.