### 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}.

### 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.

# Big Fish

##### Age 14 to 16 ShortChallenge Level

Answer: $54$ kg

Body weighs $b$ kg, head $h$ kg, tail $9$ kg

$h=9+\frac13b$ and $b=h+9$, so \begin{align}b&=h+9\\ &=\left(9+\tfrac13b\right)+9\\ &=\tfrac13b+18\\ \Rightarrow\tfrac23b&=18\\ \Rightarrow\ \ b&=27\end{align}
So $h=9+\frac13\times27=18$ and the fish weighs $27+18+9=54$ kg

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.