When studying mathematics, one often finds something like:
'The following statements are equivalent:
$1$. Statement $1$
$2$. Statement $2$
$n$. Statement $n$'
I'm looking for some examples where the implications $1\implies 2 \implies \ldots\implies n\implies 1$ are trivial or very easy but the converse implications are very hard without going through another statement, i.e. each arrow in $n\implies n-1\implies\ldots\implies 1\implies n$ is hard to prove directly.