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Diophantine N-tuples

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

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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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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.

A Tilted Square

Age 14 to 16 Challenge Level:


The original square had corners (5,5), (5,6), (6,6) and (6,5). Make some different squares.


A corner is “opposite" another corner if they have no line directly between them.

Make a square where the corners (5,3) and (5,7) are opposite each other.


A square is “tilted” if it has no vertical sides. The square you were just asked to draw is an example of a tilted square.

Make another tilted square, this one with corners (5,3) and (6,6) opposite each other.

How do you know that what you have drawn is a square?


Which of these collections of points describes a square?

A) (8,3), (7,8), (2,7), (3,2)
B) (3,3), (7,4), (8,8), (4,7)
C) (16,19), (18,22), (21,20), (19,17)
D) (4,20), (21,19), (20,2), (3,3)

What is special about sets of points that make a square?


Write down all the squares that have corners (61, 26) and (78, 43).
(Note: these do not need to be opposite).


The opposite vertices of a square have coordinates (a,b) and (c,d).

What are the coordinates of the other two vertices?