\documentclass{article}
\usepackage{amsmath}
\title{The perverse timing of stepping the equation counter}
\begin{document}
\maketitle
\section{Equations}
Note how equations increment the equation counter at the \emph{beginning},
even if later on, the equation tag is removed or set explicitly!
Normal (w/eq=\theequation)
\begin{equation}
N=\mbox{\theequation}
\end{equation}
Nonumber before (w/eq=\theequation)
\begin{equation}
\nonumber
N=\mbox{\theequation}
\end{equation}
Nonumber after (w/eq=\theequation)
\begin{equation}
N=\mbox{\theequation}
\nonumber
\end{equation}
Numbered, Tag before (w/eq=\theequation)
\begin{equation}
\tag{\theequation$'$}
N=\mbox{\theequation}
\end{equation}
Numbered, Tag after (w/eq=\theequation)
\begin{equation}
N=\mbox{\theequation}
\tag{\theequation$''$}
\end{equation}
Other variations:
\begin{equation}
N=\mbox{\theequation}
\tag{\theequation\texttt{ primed}}
\end{equation}
\begin{equation}
N=\mbox{\theequation}
\tag{\theequation\ifmmode~in math\else~in text\fi}
\end{equation}
UnNumbered (w/eq=\theequation)
\begin{equation*}
N=\mbox{\theequation}
\end{equation*}
UnNumbered, Tag before (w/eq=\theequation)
\begin{equation*}
\tag{\theequation$'''$}
N=\mbox{\theequation}
\end{equation*}
UnNumbered, Tag after (w/eq=\theequation)
\begin{equation*}
N=\mbox{\theequation}
\tag{\theequation$''''$}
\end{equation*}
Now (w/eq=\theequation).
\section{Eqnarrays}
Note how eqnarrays \emph{also} increment at the beginning!
\emph{Even} eqnarray*!
Mostly numbered, unnumbered before and after (w/eq=\theequation)
\begin{eqnarray}
N&=&\mbox{\theequation} \\
N&=&\mbox{\theequation} \\
\nonumber N&=&\mbox{\theequation} \\
N&=&\mbox{\theequation} \nonumber\\
N&=&\mbox{\theequation}
\end{eqnarray}
Unnumbered (w/eq=\theequation)
\begin{eqnarray*}
N&=&\mbox{\theequation} \\
N&=&\mbox{\theequation}
\end{eqnarray*}
Now  (w/eq=\theequation).
\section{AMS alignments}
However, note how AMS alignments \emph{defer} incrementing the equation counter
until the end of line mark!
Align numbered, some unnumbered before, after, some tagged before, after (w/eq=\theequation)
\begin{align}
N&=\mbox{\theequation} \\
N&=\mbox{\theequation} \\
\nonumber N&=\mbox{\theequation} \\
N&=\mbox{\theequation} \nonumber\\
N&=\mbox{\theequation} \\
\tag{\theequation$'$} N&=\mbox{\theequation} \\
N&=\mbox{\theequation} \tag{\theequation$''$}
\end{align}
Align numbered, some tagged before, after (w/eq=\theequation)
\begin{align*}
N&=\mbox{\theequation} \\
\tag{\theequation$'''$} N&=\mbox{\theequation} \\
N&=\mbox{\theequation} \tag{\theequation$''''$}\\
N&=\mbox{\theequation} \\
\end{align*}
Now (w/eq=\theequation).
\end{document}