exercise package; how to get counter for a custom exercise type to increment with memoir sections?

Question

Just started learning LaTeX, because I'm trying to write a short math book. I started out using LyX, but ultimately I want more control over the typesetting, so I have to learn LaTeX.

Anyway, I want the \newenvironment{Example} that I defined to update its counter in the same way that \exercise does.

\exercise updates the way I want it to when I use \counterwithin{Exercise}{section}, but it does not work when I use \counterwithin{Example}{section}. I have no idea what the problem is, any help would be greatly appreciated.

Code that illustrates my problem:

\documentclass{memoir}
\usepackage{exercise,chngcntr}


\begin{document}

%Memoir
\setcounter{secnumdepth}{2}


\chapter{Chapter name}
\section{Section1}

\newcounter{Example}

\counterwithin{Exercise}{section} %Ties exercises counter to memoir section?
\counterwithin{Example}{section} %Use this to tie the example counter to the memoir sections?

\newenvironment{Example}{\begin{Exercise}[name={Example},
counter={Example}]}
{\end{Exercise}}

%Exercises below:

\begin{Exercise}
Exercise text
\end{Exercise}

\begin{Example}
Example text
\end{Example}

\begin{Exercise} %counter is the way I want it to be
Exercise text
\end{Exercise}

\section{Section2}

\begin{Exercise}
Exercise text
\end{Exercise}

\begin{Example}
Example text
\end{Example}

\begin{Example} %Counter does not increment
Example text
\end{Example}

\begin{Exercise}
Exercise text
\end{Exercise}

\end{document}

Edit: For clarification, the problem is in the second section, Section2, where the Example doesn't update: Example 2.1.1 and then Example 2.1.1 again, instead of Example 2.1.1 and then Example 2.1.2.

Answer 1

The exercise package offers the command \renewcounter{<counter>}[<resetcounter>] to modify the behaviour of predefined counters.

So you need to add \renewcounter{Exercise}[section] to your preamble.