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Cosines Rule

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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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Root to Poly

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

Orbiting Billiard Balls

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

This is a hard but hopefully interesting scenario.

Drawing out the table(s) and experimenting with different paths (angles) will generate a 'feel' for the problem and begin to suggest some features and underlying relationships.

Ideally the context is rich in possibilities for extension and fresh questions. For example are orbital paths possible where a rebound does not always take the ball on to the adjacent side but instead across to the opposite side, and if such paths exist, what is the connection between simple orbits and those where the cycle involves more than 4 rebounds?