% https://tex.stackexchange.com/q/576998
\documentclass[twocolumn]{book}
\usepackage{xsim}

\DeclareExerciseEnvironmentTemplate{number-only}{%
  \par\noindent
  \textbf{\GetExerciseProperty{counter}}%
  \GetExercisePropertyT{subtitle}{ \textit{#1}} %
}{\par}

\newcommand\printsectionexercises{%
  \ForEachUsedExerciseByType{%
    \ifnum\ExercisePropertyGet{##1}{##2}{chapter-value}=\value{chapter}
      \ifnum\ExercisePropertyGet{##1}{##2}{section-value}=\value{section}
        \XSIMprint{exercise}{##1}{##2}%
      \fi
    \fi
  }%
}

\renewcommand\printsolutions{%
  \def\currentchapter{}%
  \def\currentsection{}%
  \def\lastchapter{}%
  \def\lastsection{}%
  \ForEachUsedExerciseByType{%
    \let\lastchapter\currentchapter
    \let\lastsection\currentsection
    \edef\currentchapter{\ExercisePropertyGet{##1}{##2}{chapter-value}}%
    \edef\currentsection{\ExercisePropertyGet{##1}{##2}{section-value}}%
    \ifx\lastchapter\currentchapter\else
      \section*{Chapter \ExercisePropertyGet{##1}{##2}{chapter}}
    \fi
    \ifx\lastsection\currentsection\else
      \subsection*{Exercises \ExercisePropertyGet{##1}{##2}{section}}
    \fi
    \XSIMprint{solution}{##1}{##2}%
  }%
}

\xsimsetup{
  exercise/template = number-only ,
  solution/template = number-only ,
  exercise/within = section ,
  exercise/the-counter = \arabic{exercise}. ,
  exercise/print = false
}

\begin{document}

\setcounter{chapter}{13}
\chapter{}
\section{Cats as pets}
Text text text\dots
\begin{exercise}
  Draw a cat.
\end{exercise}
\begin{solution}
  Drawing a cat.
\end{solution}

Text text text\dots
\begin{exercise}
  Bathe a cat.
\end{exercise}
\begin{solution}
  Bathing a cat.
\end{solution}

Text text text\dots
\begin{exercise}
  Trace the history of cats through time.
\end{exercise}
\begin{solution}
  Tracing the history of cats through time.
\end{solution}

\subsection*{Exercises \thesection}
\printsectionexercises

\section{Frogs as pets}
Text text text\dots
\begin{exercise}
  Lick a frog.
\end{exercise}
\begin{solution}
  Licking a frog.
\end{solution}

Text text text\dots
\begin{exercise}
  Bathe a frog.
\end{exercise}
\begin{solution}
  Bathing a frog.
\end{solution}

Text text text\dots
\begin{exercise}
  Feed a frog.
\end{exercise}
\begin{solution}
  Feeding a frog.
\end{solution}

\subsection*{Exercises \thesection}
\printsectionexercises

\setcounter{chapter}{20}
\chapter{}

\section{The joys of fish}
Text text text\dots
\begin{exercise}
  Catch a fish.
\end{exercise}
\begin{solution}
  Catching a fish.
\end{solution}

Text text text\dots
\begin{exercise}
  Draw a fish.
\end{exercise}
\begin{solution}
  Drawing a fish.
\end{solution}

\subsection*{Exercises \thesection}
\printsectionexercises

\chapter*{Solutions to the exercises}
\printsolutions

\end{document}