There is a very interesting puzzle for Fibonacci sequence:
You are climbing a stair case. It takes n steps to reach to the top. Each time you can either climb $1$ or $2$ steps. In how many distinct ways can you climb to the top?
The answer is Fibonacci sequence: $F(n) = F(n - 1) + F(n - 2)$. The explanation is that you could reach the $n$th step either from the $(n - 1)$th step or $(n - 2)$th step.
What I fail to understand is why isn't the answer $F(n) = (F(n - 1) + 1) + (F(n - 2) + 2)$. Since, after reaching $(n - 1)$th step I have to take $1$ step to reach the $n$th step and after reaching $(n - 2)$th step I have to take another $2$ steps to reach the $n$th step.
I know I might be ignoring some very stupid detail but it's better to ask and be called a fool once than to remain a fool forever.