Number invariant problem: replacing any two numbers $a$ and $b$ with $a - 1$ and $b + 3$ 
Numbers 1, 2, 3, ..., 2014 are written on a blackboard. Every now and then somebody picks two numbers $a$ and $b$ and replaces them by $a - 1$, $b + 3$. Is it possible that at some point all numbers on the blackboard are even? Can they all be odd?

I cannot figure out how to use the terms $a - 1$ and $b  + 3$ to form a invariant.

Hint: subtracting $1$ or adding $3$ changes a number's parity (i.e. it goes from odd to even or vice versa).  When you do this to two numbers, the number of odd numbers on the blackboard  changes by either $-2$, $0$ or $+2$.
