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There's Always One Isn't There
Age 14 to 16
Challenge Level 
Systematic working and recording of results help a lot here.
Conjectures are important, and should be encouraged, but along with a challenge to really explain why any claim might be true generally.
We are so familiar with numbers and what they do, or what we believe they do, that the challenge to account rigorously for the familiar can seem pedantic. Hopefully the problem expressed in this form will give students the pleasure of discerning real structure.
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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}.
