in a school, there are teachers and students; students are hot; they have many, many exams to take; if they do well in an exam, they get an a; if not, an f; in either case, they receive a grade sheet from their teachers; if they take an f grade sheet back home, they can redeem it for a free beat at their moms; guess, what, hot moms;

nevertheless, teachers are hot, too; many teachers typeset documents using latex templates; for example, a texsavvy teacher can use a template to produce a grade sheet for students:

\documentclass{article}
\def\b{\relax\relax}
\def\g{\relax}
\edef\f{\b}
\def\grade#1{\edef\f{#1}\ifx\f\g A\else F\fi}
\begin{document}
\tableofcontents
\clearpage
\section{You got an \grade{\f} in the test!}
\end{document}

this template contains only 10 lines of code, but does its job: it first defines a “bad” token \b and a “good” token \g, then sets a flag \f to one of \b and \g, depending on how well the student performed in the exam; finally, the macro \grade turns the flag into a grade: either A or F, for print out;

compile this document (twice) and the output looks like this:

F.png

if the student did well in the exam, the teacher can change:

\edef\f{\b}

into:

\edef\f{\g}

and the output looks like this:

A.png

the definition of macro \f is the only thing in the template that the teacher may change; and the output always has 2 pages: first page has table of contents, second page has section title;

a failed student

if a student failed the test and got an f grade sheet, he probably does not want his mom to know about it; unfortunately, the teacher will split the grade sheet, and mail the first page to student parent for review; since the teacher believes the table of contents lists the same section title, only the second page is kept local to the teacher;

so, the life of the student is tough: he must let his teacher see an F and his mom an A; both his teacher and his mom are very hot; if he was caught cheating a hot beat is unavoidable; fortunately, the student himself is also very hot: he found a way to tamper with the latex template in a limited fashion:

  • the student must not delete or comment out existing code;

  • the student must not redefine existing macros except \f;

  • the student must not define or redefine more than 2 macros;

question

is it possible for the student to avoid a beat? this means, he must modify the template so that its output prints an A on the first page, but an F on the second page;

analysis

what a hot question!

at first glance, it requires the document to output different texts in the table of contents and section title; but should not they always be the same? well well there is a well or not well known difference between should be and could be; while they should be the same, they could be different;

when a latex document has a toc, it needs to be compiled twice to update the toc correctly; latex works sequentially, so it doesnt know beforhand about sections; the best it can do, is to collect sections in the first run, then create the toc using collected data, in the second run; the collected data are stored in .aux file, which is used to generate .toc file, which contains toc entries; to know more, read this thread;

now we can peek into .toc file to see what is generated; here is its content:

\contentsline {section}{\numberline {1}You got an \edef \relax \relax {\relax \relax }F in the test!}{2}%

note that \edef \relax \relax {\relax \relax } defines \relax with parameter text \relax and outputs nothing; so \grade{\f} is transformed into F; note that the definition of \grade{\f} is \edef\f{#1}\ifx\f\g A\else F\fi; how is this expanded into \edef \relax \relax {\relax \relax }F (and so F)?

answer:

\edef                => \edef
\f                   => \relax\relax
{                    => {
#1                   => \relax\relax
}                    => }
\ifx\f\g A\else F\fi => F

now our job is to change the expansion of \edef\f{#1}\ifx\f\g A\else F\fi into A; since we can change the definition of \f, the result will be like this:

\edef<something>{<something>}\ifx<something>\g A\else F\fi

where <something> is the definition of \f;

the tricky thing is: how can we generate different results in table of contents and in section title using the same command sequence? to answer this, note that there is a subtle difference as for how these two parts are generated:

  • section title is generated in the first pass of compilation, while expanding and executing the command sequence;

  • table of contents is generated in the second pass of compilation, using data generated in the first pass of compilation (.toc file); however, this data is generated by expanding but not executing the command sequence; this has been shown above;

it is clear that we cannot do much with the section title: no matter how \f is defined, \edef\f{#1} outputs nothing; in fact there are only two possibilities about the definition of \f: if \f is defined as \relax, then \ifx\f\g is true and the whole command sequence expands into A; else, F; here it must be F because this page will be validated by the teacher; therefore \f cannot be defined as \relax;

now our job is to define \f so that the same command sequence expands into A when expanded without execution; the first obstacle is the beginning \edef: we cannot do much if the whole \edef command has \edef<something>{<something>}; that would leave us with \ifx<something>\g A\else F\fi where we cant really do a lot; fortunately, we can terminate this \edef early, with a clever choice of <something>, for example: \relax{}; when we put \relax{} into this command sequence, what we get is:

\edef\relax{}{\relax{}}\ifx\relax{}\g A\else F\fi

this still outputs F; however dont forget that we can do more than \relax{}; specifically, we can put a pacman after \relax{}: a hot pacman that eats tokens ahead; such pacman is easy to define; for example:

\def\p#1#2#3#4{}

this pacman eats 4 tokens ahead and outputs nothing;

in our case, we only need to eat 2 more tokens:

  • {<something>};

  • \ifx<something>\g A\else F\fi; this is F when <something> is not \relax (the definition of \g);

so our pacman is:

\def\p#1#2{}

with this pacman, the definition of \f becomes:

\def\f{\relax{}\p}

and the whole command sequence is:

\edef\relax{}\p{\relax{}\p}F

however, when we test this, it doesnt compile:

! Argument of \p has an extra }.

it is not obvious from the error message where it goes wrong; actually the error comes from this:

  • when we execute \grade{\f}, the tex engine executes \edef\f{\f} because this is part of the definition of \grade and here #1 is \f;

  • \edef\f{\f} will expand \f, which is defined as \relax{}\p;

  • \p needs 2 tokens; but what follows (the second) \f in \edef\f{\f} is a brace };

to fix this, we need to tell the tex engine not to expand \p; this can be done with \protected:

\protected\def\p#1#2{}

now if we compile the code twice, the .toc file contains:

\contentsline {section}{\numberline {1}You got an \edef \relax {}\p {\relax {}\p }F in the test!}{2}%

now if we look at the .pdf file, it is very close to what we want; we need one more small change to the pacman: output an A:

\protected\def\p#1#2{A}

the .toc file remains the same because \p is protected; and the .pdf file is exactly what we want:

AF.png

solution

the solution is so hot: only 2 more lines of code:

\documentclass{article}
\def\b{\relax\relax}
\def\g{\relax}
\edef\f{\b}
\protected\def\p#1#2{A}     %%  added;
\def\f{\relax{}\p}          %%  added;
\def\grade#1{\edef\f{#1}\ifx\f\g A\else F\fi}
\begin{document}
\tableofcontents
\clearpage
\section{You got an \grade{\f} in the test!}
\end{document}

this should be hot enough to help the hot student avoid a hot beat; however, if the hot teacher knows to protect the hot macro \grade, then …