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Diophantine N-tuples

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

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}.


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.

Chocolate 2010

Age 14 to 16
Challenge Level

ChocolateFirst of all, pick the number of times a week that you would like to eat chocolate (try for more than once but less than 10 times as more than this is simply greedy).
Multiply this number by 2 (just to be bold).
Add 5 (for Sunday).
Multiply it by 50.
If you have already had your birthday this year add 1760.
If you haven't had your birthday yet this year add 1759.
Now subtract the four digit year that you were born.
You should have a three digit number.

The first digit of this was your original number.
The next two numbers are your age.

This problem was first published in 2010. How does this work? Does this work for any year? If not, can you adapt it so that it does?

Thanks to Jose Luis for the idea for this question.