### 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}.

### Common Divisor

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

### Hypotenuse Lattice Points

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

# Different by One

##### Stage: 4 Challenge Level:
Take two rod colours, say light green (3 units) and yellow (5 units) - as many as you like of each colour.

Make a line of greens and a line of yellows so that the lines differ in length by one (a white rod)

Now try with other colour combinations. For example blue with purple - you'll want to try lots of combinations.

Keep a record of your results.

What do you notice ? Why does it happen ?

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The longest rod (orange) is 10 units long but we do not have to stop at 10. You can have a blue-yellow rods (length 14 units) and orange-black rods (17 units long) for example.

Six blue-yellow rods and five orange-black rods make lines that differ by one (white)

Once you think you have spotted what happens for a difference of 1, try a difference of 2, using plenty of different colour combinations.

Especially notice the cases where a difference of 2 can be achieved but a difference of 1 cannot.

Record, notice and explain what happens.