### Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares?

### DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

### Sixational

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

# 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.