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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}.

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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.

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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

Age 14 to 16 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)

Yellow line and Green line differ by one

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.

What about other differences ?