Generating random numbers without repetitions

Question

How do I generate in LaTeX a list of random integers, in a given range, avoiding repetitions? The command \pgfmathrandomitem selects an item from a list, but when inserted in a loop can select twice the same item. Since I am selecting questions from a list in order to prepare an exam, I cannot ask twice the same question.

Answer 1

You can remove the item from the list after selecting it, then it won't be picked again:

\documentclass{article}
\usepackage{tikz}

\begin{document}
\makeatletter
\pgfmathsetseed{123321}



\def\prunelist#1{%
\expandafter\edef\csname pgfmath@randomlist@#1\endcsname
        {\the\numexpr\csname pgfmath@randomlist@#1\endcsname-1\relax}
\count@\pgfmath@randomtemp 
\loop
\expandafter\let
\csname pgfmath@randomlist@#1@\the\count@\expandafter\endcsname
\csname pgfmath@randomlist@#1@\the\numexpr\count@+1\relax\endcsname
\ifnum\count@<\csname pgfmath@randomlist@#1\endcsname\relax
\advance\count@\@ne
\repeat}

\pgfmathdeclarerandomlist{mylist}{{one}{two}{three}{four}{five}{six}{seven}}

\pgfmathrandomitem\z{mylist}\z\prunelist{mylist}


\pgfmathrandomitem\z{mylist}\z\prunelist{mylist}

\pgfmathrandomitem\z{mylist}\z\prunelist{mylist}

\pgfmathrandomitem\z{mylist}\z\prunelist{mylist}

\pgfmathrandomitem\z{mylist}\z\prunelist{mylist}

\pgfmathrandomitem\z{mylist}\z\prunelist{mylist}

\pgfmathrandomitem\z{mylist}\z\prunelist{mylist}



\end{document}

---

to declare an integer list:

\def\declarenumlist#1#2#3{%
\expandafter\edef\csname pgfmath@randomlist@#1\endcsname{#3}%
\count@\@ne
\loop
\expandafter\edef
\csname pgfmath@randomlist@#1@\the\count@\endcsname
  {\the\count@}
\ifnum\count@<#3\relax
\advance\count@\@ne
\repeat}

%\pgfmathdeclarerandomlist{mylist}{{one}{two}{three}{four}{five}{six}{seven}}


\declarenumlist{mylist}{1}{10}% list from 1 to 10 inclusive.

Answer 2

This code for generating random numbers from 1 to N without repetitions should run faster than the one based on pgfmath. Speed gain can not be felt here because the numbers are extracted one by one, but if one wanted to generate in one-go a full random permutation, for later use, then this code could be extended to do it and the difference with the pgfmath+\prune should show up when dealing with N more than a few hundreds. (untested ;-) ).

Edit: code added to generate in one-go a complete random permutation of the integers from 1..N. Contrarily to \NewRandom which each time "prunes" the list by one, \NewPermutation leaves the list intact in order to allow iterated uses.

Last Edit: added \DeclareList as a companion to \DeclareIntegerList, but now for lists of arbitrary things. Renamed \NewPermutation to \Permutation and rewrote the code which does not re-use anymore pieces of \NewRandom with its accompanying overhead.

\documentclass{article}
\usepackage[vscale=0.9,hscale=0.75]{geometry}

% \DeclareIntegerList {name}{N}
% declares the list of the integers from 1 to N 
% under name "name" for future use of either
% \NewRandom<macro>\FromList {name}, 
% or \Permutation<macro>\OfList{name}

% \DeclareList {name}{{first}{second}...{last}}
% declares similarly a list of things, possibly containing empty lines or
% `\par`. Second argument may be macro. Items must be braced (or single tokens
% but the first one will be expanded if not protected by braces)

% \NewRandom\z \FromList {name}

% makes \z expand to a randomly chosen element in what remains of list
% "name", and removes the chosen element from the list "name"

% \Permutation\R \OfList {name}

% makes \R expand to the token list of the braced elements of list
% "name", in a random order. The list is not modified.

\makeatletter

\newtoks\rndperm@tok
\newcount\rndperm@cnta
\newcount\rndperm@cntb

% auxiliary macros \rndperm@tmpa, \rndperm@tmpb, \rndperm@aux

\newcommand{\DeclareIntegerList}[2]{%
% sets up a the integers from 1 to N as a list for random extraction
    \rndperm@cnta #2\relax
    \rndperm@cntb \rndperm@cnta
    \rndperm@tok {}%
    \loop
    \rndperm@tok 
             \expandafter\expandafter\expandafter{%
             \expandafter\the\expandafter\rndperm@tok
             \csname \the\rndperm@cntb\expandafter\endcsname 
                   \expandafter{\the\rndperm@cntb}}%
    \ifnum\rndperm@cntb>\@ne
       \advance\rndperm@cntb\m@ne
    \repeat
    \expandafter\def\csname rndperm@list@#1\expandafter\endcsname
    \expandafter{\the\rndperm@tok }%
    \expandafter\def\csname rndperm@card@#1\expandafter\endcsname
    \expandafter{\the\rndperm@cnta }%
}%

\newcommand{\NumberOfItems}[1]{\csname rndperm@card@#1\endcsname }

\newcommand{\DeclareList}[2]{%
    \rndperm@cnta \z@
    \rndperm@tok {}%
    \expandafter\rndperm@scanlist \romannumeral-`0#2\rndperm@bye
    \long\expandafter\def\csname rndperm@list@#1\expandafter\endcsname
         \expandafter{\the\rndperm@tok }%
    \expandafter\def\csname rndperm@card@#1\expandafter\endcsname
                \expandafter{\the\rndperm@cnta }%
}%

\long\def\rndperm@bye #1\rndperm@bye {}%
\def\rndperm@endscan #1\rndperm@scanlist {}%

\long\def\rndperm@scanlist #1{%
     \rndperm@bye #1\rndperm@endscan\rndperm@bye
     \advance\rndperm@cnta\@ne
     \long\expandafter\def\expandafter\rndperm@tmpa\expandafter
          {\csname\the\rndperm@cnta\endcsname {#1}}%
     \rndperm@tok\expandafter\expandafter\expandafter
                {\expandafter\rndperm@tmpa\the\rndperm@tok }%
     \rndperm@scanlist
}%


\def\NewRandom #1\FromList #2{%
    \rndperm@cnta \csname rndperm@card@#2\endcsname\relax
    \ifnum\rndperm@cnta=\z@\rndperm@abort #1\fi
    \expandafter\let\expandafter\rndperm@list\csname rndperm@list@#2\endcsname
    \rndperm@cntb\pdfuniformdeviate\rndperm@cnta\relax
    \advance\rndperm@cnta\m@ne
    \ifnum\rndperm@cntb=\rndperm@cnta
            \expandafter\rndperm@firstone
       \else
            \expandafter\rndperm@fartherup
    \fi
    \expandafter\def\csname rndperm@card@#2\expandafter\endcsname 
                \expandafter{\the\rndperm@cnta}%
    \long\expandafter\def\expandafter#1\expandafter{\rndperm@element}%
    \expandafter\let\csname rndperm@list@#2\endcsname\rndperm@list
     \empty
}%

\def\rndperm@abort #1\fi #2\empty{\fi \let#1\empty}

\def\rndperm@firstone {\expandafter\rndperm@firstone@a \rndperm@list!}%

\long\def\rndperm@firstone@a #1#2#3!{%
     \long\def\rndperm@element {#2}%
     \long\def\rndperm@list {#3}}

\def\rndperm@fartherup {%
    \expandafter\def\expandafter\rndperm@tmpa
                \expandafter{\csname\the\numexpr\rndperm@cntb+1\endcsname }%
    \rndperm@tok {\long\def\rndperm@aux ##1##2##3}%
    \expandafter\the\expandafter\rndperm@tok\rndperm@tmpa ##4##5!%
       {\long\expandafter\def
        \expandafter\rndperm@tmpa\expandafter{\rndperm@tmpa {##2}##5}%
        \long\def\rndperm@tmpb {##3}%
        \expandafter\expandafter\expandafter\def
        \expandafter\expandafter\expandafter\rndperm@list
        \expandafter\expandafter\expandafter
                    {\expandafter\rndperm@tmpb\rndperm@tmpa }%
        \long\def\rndperm@element {##4}}%
     \expandafter\rndperm@aux \rndperm@list!%
}%

\def\Permutation #1\OfList #2{%
    \rndperm@cnta \csname rndperm@card@#2\endcsname\relax
    \let\rndperm@perm\empty
    \ifnum\rndperm@cnta>\z@ 
            \expandafter\expandafter\expandafter
            \rndperm@all\csname rndperm@list@#2\endcsname
    \fi
    \let#1\rndperm@perm
}%

\def\rndperm@all {%
    \rndperm@cntb\pdfuniformdeviate\rndperm@cnta\relax
    \advance\rndperm@cnta\m@ne
    \ifnum\rndperm@cntb=\rndperm@cnta
            \expandafter\rndperm@all@firstone
       \else
            \expandafter\rndperm@all@fartherup
    \fi
}%

\long\def\rndperm@all@firstone #1#2{%
    \long\expandafter\def\expandafter\rndperm@perm\expandafter
                               {\rndperm@perm {#2}}%
    \ifnum\rndperm@cnta>\z@ \expandafter\rndperm@all\fi 
}%

\def\rndperm@all@fartherup {%
    \expandafter\def\expandafter\rndperm@tmpa
                \expandafter{\csname\the\numexpr\rndperm@cntb+1\endcsname }%
    \rndperm@tok {\long\def\rndperm@aux ##1##2##3}%
    \expandafter\the\expandafter\rndperm@tok\rndperm@tmpa ##4%
       {\long\expandafter\def\expandafter\rndperm@perm\expandafter
                               {\rndperm@perm {##4}}%
        \long\expandafter\def\expandafter\rndperm@tmpa
                        \expandafter{\rndperm@tmpa {##2}}%
        \long\def\rndperm@tmpb {##3}%
        \expandafter\expandafter\expandafter\rndperm@all
        \expandafter\rndperm@tmpb\rndperm@tmpa 
       }%
     \rndperm@aux 
}%

\makeatother

\begin{document}
\begin{verbatim}
\DeclareIntegerList {mylist}{10}
\count 255 1
\loop
   \NewRandom\z\FromList {mylist}\z 
\ifnum\count 255 < 15 \advance\count 255 by 1 ,
\repeat 
\end{verbatim}

\DeclareIntegerList {mylist}{10}
\count 255 1
\loop
   \NewRandom\z\FromList {mylist}\z 
\ifnum\count 255 < 15 \advance\count 255 by 1 ,
\repeat (macro is set to empty when the list is exhausted)

\begin{verbatim}
\DeclareList {words}{{Unpounced}{epimetheus}{angelhood}{diopside}% (percent optional)
                      {gospeler}{quiff}{eutrophy}}
\count 255 1
{\bfseries\loop
   \NewRandom\z\FromList {words}\z 
\ifnum\count 255 < 7 \advance\count 255 by 1 ,
\repeat } 
\end{verbatim}

\DeclareList {words}{{Unpounced}{epimetheus}{angelhood}{diopside}%
                      {gospeler}{quiff}{eutrophy}}
\count 255 1
{\bfseries\loop
   \NewRandom\z\FromList {words}\z 
\ifnum\count 255 < 7 \advance\count 255 by 1 ,
\repeat }

At this stage \texttt{words} has no more elements, it is emptied.


% \verb|\Permutation\S\OfList {mylist}| sets \texttt{\string\S} to the empty macro
% (meaning of \texttt{\string\S} is \Permutation\S\OfList
% {mylist}\texttt{\meaning\S}).

\begin{verbatim}
\def\x #1{\ifx \relax #1\else \{#1\}\hskip 0pt plus 1pt minus 1pt
                       \expandafter\x \fi }
\DeclareList {words}{{Unpounced}{epimetheus}{angelhood}{diopside}%
                     {gospeler}{quiff}{eutrophy}}
\Permutation\S\OfList {words}\texttt{\string\S\ is \expandafter\x\S\relax}\endgraf
\end{verbatim}

\def\x #1{\ifx \relax #1\else \{#1\}\hskip 0pt plus 1pt minus 1pt
                       \expandafter\x \fi }
\DeclareList {words}{{Unpounced}{epimetheus}{angelhood}{diopside}%
                     {gospeler}{quiff}{eutrophy}}
\noindent
\Permutation\S\OfList {words}\texttt{\string\S\ is \expandafter\x\S\relax}\endgraf
\noindent\Permutation\S\OfList {words}\texttt{\string\S\ is \expandafter\x\S\relax}\endgraf
\noindent\Permutation\S\OfList {words}\texttt{\string\S\ is \expandafter\x\S\relax}\endgraf
\noindent\Permutation\S\OfList {words}\texttt{\string\S\ is \expandafter\x\S\relax}\endgraf

\medskip
We can declare lists of arbitrary contents
\begin{verbatim}
\DeclareList {tokens}{{\if}{{\ifnum}}{\ifdim}{\par}{\end}{\fi}{{\fi\fi}}{\gobbleall}}%
There are \NumberOfItems {tokens} elements in this list.\endgraf
\noindent\Permutation\S\OfList {tokens}\noindent\texttt{\meaning\S}\endgraf
\noindent\Permutation\S\OfList {tokens}\noindent\texttt{\meaning\S}\endgraf
\noindent\Permutation\S\OfList {tokens}\noindent\texttt{\meaning\S}\endgraf
\noindent\Permutation\S\OfList {tokens}\noindent\texttt{\meaning\S}\endgraf
\end{verbatim}

\DeclareList {tokens}{{\if}{{\ifnum}}{\ifdim}{\par}{\end}{\fi}{{\fi\fi}}{\gobbleall}}
There are \NumberOfItems {tokens} elements in this list.\endgraf
\noindent\Permutation\S\OfList {tokens}\texttt{\meaning\S}\endgraf
\noindent\Permutation\S\OfList {tokens}\texttt{\meaning\S}\endgraf
\noindent\Permutation\S\OfList {tokens}\texttt{\meaning\S}\endgraf
\noindent\Permutation\S\OfList {tokens}\texttt{\meaning\S}\endgraf

\begin{verbatim}
\DeclareIntegerList {mylist}{200}
\texttt{mylist} has \NumberOfItems {mylist} elements and a random element is
\NewRandom\z \FromList {mylist}\texttt{\z}. There are now only \NumberOfItems
{mylist} elements left. Here they are in random order:
\Permutation\S\OfList {mylist}%
\texttt{\expandafter\x\S\relax}
\end{verbatim}

\DeclareIntegerList {mylist}{200}
\texttt{mylist} has \NumberOfItems {mylist} elements and a random element is
\NewRandom\z \FromList {mylist}\texttt{\z}. There are now only \NumberOfItems
{mylist} elements left. Here they are in random order:
\Permutation\S\OfList {mylist}%
\texttt{\expandafter\x\S\relax}

\end{document}
\medskip
Extracting 200 times with no repetitions
after \verb|\DeclareIntegerList {mylist}{200}|:

\DeclareIntegerList {mylist}{200}\count 255 1
\begin{verbatim}
\count255 0
\loop
   \NewRandom\z\FromList {mylist}\z
\ifnum\count 255 < 200 \advance\count 255 1 ,
\repeat
\end{verbatim}
\count255 0
\loop
   \NewRandom\z\FromList {mylist}\z
\ifnum\count 255 < 200
\advance\count 255 1
,
\repeat

\medskip
Now with  
\begin{verbatim}
\DeclareIntegerList {mylist}{200}
\Permutation\S\OfList {mylist}
\end{verbatim}
\DeclareIntegerList {mylist}{200}
\Permutation\S\OfList {mylist}

Here are the contents of \texttt{\string\S}:
\def\x #1{\ifx \relax #1\else \{#1\}\hskip 0pt plus 1pt minus 1pt
                       \expandafter\x \fi }
\texttt{\expandafter\x\S\relax}

And \verb|\Permutation| can be used again and again to generate more random
permutations (not necessarily distinct).
% Tools of \verb|xint| such as \verb|\xintApplyUnbraced| or \verb|\xintFor*| can
% help if one wants to do things with the generated list of braced things.
\end{document}

Answer 3

Some time ago we found this code on the internet:

\input random

\newcount\icount
\newcount\i
\newcount\j

% Define a new item.
\def\defitem{%
  \advance\icount by1
  % Define a macro with the name `n' where n is the item's number.
  \expandafter\def \csname \number\icount \endcsname
}
% Printing of items.
\def\printitem#1{\csname \number#1\endcsname}
\let\printbetweenitems\space
% Print all defined items in random order.
\def\getitems{%
  % Unset all flags.
  \i=\icount
  \loop
    \expandafter\let \csname flag\number\i \endcsname a%
    \advance\i by-1
    \ifnum\i > 0 \repeat
  % Print random items, each item once, until every item has been
  % printed.
  \i=\icount
  \loop
    % Get random number \j.
    \setrannum{\j}{1}{\icount}%
    % Print item \j only if its flag is unset.
    \expandafter\ifx \csname flag\number\j \endcsname a%
      \expandafter\let \csname flag\number\j \endcsname b% Set the flag.
      \printitem\j
      \advance\i by-1
      \ifnum\i > 0 \printbetweenitems\fi
    \fi
    \ifnum\i > 0 \repeat
}


\defitem{1}
\defitem{2}
\defitem{3}
\defitem{4}
\defitem{5}
\defitem{6}
\defitem{7}
\defitem{8}
\defitem{9}
\defitem{10}
\defitem{11}

\getitems\par \getitems\par \getitems\par \getitems\par
\getitems\par

\bye

We have adapted it to create a command which chooses randomly m objects in a set of n of them (m <= n).

You can have a look to the esami.sty file at this link https://www.dropbox.com/sh/0t92kehukgafni5/0Mi0qsYLlR