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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.

2-digit Square

Age 14 to 16 Challenge Level:
Thank you Abdul for this solution.

The 2-digit number is either $65$ or $56$.

Proof:

Any 2-digit number can be represented as $10a + b$. We need $(10a+b)^2 - (10b+a)^2 = 99a^2 - 99b^2 =9 \times 11 \times (a^2 - b^2)$ to be a square.

This means that $(a^2 - b^2)$ must be 11 and so $(a - b)(a + b) = 11$ making, $a - b = 1$ and $a + b = 11$. This gives $a = 6$, and $b = 5$.

If we find a solution with $a > b$ then, by reversing the digits, we get a second solution.