How to start the equation numbering in the new box with $1$ again?

Question

\documentclass[12pt, a4paper, UTF8, scheme = plain]{ctexrep}
\usepackage{amsmath,tasks}
\usepackage{graphicx}
\usepackage{tcolorbox}
\tcbuselibrary{skins,xparse,breakable}
\tcbset{%
    colback      = white,
    colframe     = black,
    title filled = false,
    colbacktitle = white,
    breakable,
    enhanced
}%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
\NewTColorBox[
    auto counter,
%   number within    = tcolorbox,         %section ? chapter ? tcolorbox?
    number freestyle = {Example \,\noexpand\arabic{\tcbcounter} }
    ]{example}{ O{}mo }{
    fonttitle       = \bfseries,
    coltitle        = black,
    title           = {\thetcbcounter: #2},
    #1
}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{example}{box 1}
    \begin{equation}
    a + b = c
    \end{equation}
    \begin{equation}
    a + b = c
    \end{equation}
    \begin{equation}
    a + b = c
    \end{equation}
\end{example}
How to start the equation numbering in the new box with $1$ again?
\begin{example}{box 2}
    \begin{equation}
    a^2 + b^2 = c^2
    \end{equation}
    \begin{equation}
    a^3 + b^3 = c^3
    \end{equation}
    \begin{equation}
    a + b = c
    \end{equation}
\end{example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Is there an option for this in number within? I checked the manual but there was very little about it...

Answer 1

Reset the equation counter to 0 in the title of the \NewTColorBox.

\documentclass[12pt, a4paper, UTF8, scheme = plain]{ctexrep}
\usepackage{amsmath,tasks}
\usepackage{graphicx}
\usepackage{tcolorbox}
\tcbuselibrary{skins,xparse,breakable}
\tcbset{%
    colback      = white,
    colframe     = black,
    title filled = false,
    colbacktitle = white,
    breakable,
    enhanced
}%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
\NewTColorBox[
    auto counter,
%   number within    = tcolorbox,         %section ? chapter ? tcolorbox?
    number freestyle = {Example \,\noexpand\arabic{\tcbcounter} }
    ]{example}{ O{}mo }{
    fonttitle       = \bfseries,
    coltitle        = black,
    title           = {\thetcbcounter: #2\setcounter{equation}{0}},
    #1
}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{example}{box 1}
    \begin{equation}
    a + b = c
    \end{equation}
    \begin{equation}
    a + b = c
    \end{equation}
    \begin{equation}
    a + b = c
    \end{equation}
\end{example}
How to start the equation numbering in the new box with $1$ again?
\begin{example}{box 2}
    \begin{equation}
    a^2 + b^2 = c^2
    \end{equation}
    \begin{equation}
    a^3 + b^3 = c^3
    \end{equation}
    \begin{equation}
    a + b = c
    \end{equation}
\end{example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}%%%%%%%%%%

Answer 2

You should define a special counters for equations inside the boxes. It's not difficult to replace the equation counter with the special counter, that's reset at each example environment.

I also add support for hyperref.

\documentclass{article}
\usepackage{amsmath,tasks}
\usepackage{graphicx}
\usepackage{tcolorbox}
\tcbuselibrary{skins,xparse,breakable}

\usepackage{hyperref}

\tcbset{
  colback      = white,
  colframe     = black,
  title filled = false,
  colbacktitle = white,
  breakable,
  enhanced
}
\NewTColorBox[
  auto counter,
  number freestyle = {Example \noexpand\arabic{\tcbcounter}}
]{example}{ O{}mo }{
  fonttitle = \bfseries,
  coltitle  = black,
  title     = {\thetcbcounter: #2},
  #1
}

\newcounter{boxequation}
\AddToHook{env/example/begin}{%
  \setcounter{boxequation}{0}%
  \ExpandArgs{cc}\let{c@equation}{c@boxequation}%
  % comment the following line if you're not using hyperref
  \renewcommand{\theHequation}{\arabic{\tcbcounter}\arabic{boxequation}}%
}

\begin{document}

\begin{example}{box 1}
Some text about this example
    \begin{gather}
    a + b = c \\
    a + b = c \\
    a + b = c
    \end{gather}
Some text about this example
\end{example}
How to start the equation numbering in the new box with $1$ again?
\begin{example}{box 2}
Some text about this example
    \begin{gather}
    a^2 + b^2 = c^2 \\
    a^3 + b^3 = c^3 \\
    a + b = c
    \end{gather}
Some text about this example
\end{example}

\end{document}