I found an astonishing result in a problem today here !
A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before.
$44 \to 32 \to 13 \to 10 \to 1 \to 1$
$85 \to 89 \to 145 \to 42 \to 20 \to 4 \to 16 \to 37 \to 58 \to 89$
Therefore any chain that arrives at $1$ or $89$ will become stuck in an endless loop. What is most amazing is that EVERY starting number will eventually arrive at $1$ or $89$.
Why is this true?