Command Defining Nested Command

Question

I'm trying to define a command which, upon being called with a certain parameter, 'renews' a different command. In particular, I tried the following (it doesn't work!).

\newcommand{\gap} %initialise \gap to exist
\newcommand{\GAP}{1}{
    \renewcommand{\gap}{\hspace{#1 em}} }

My idea was that \GAP{3} redefines \gap to \hspace{3em}, for example.

I've looked at a few TeX.SE questions, in particular this one, but I can't see how to correct this...

Incidentally, removing the space between #1 and em doesn't work, and putting two #-s outputs 13 (for \GAP{3}).

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Here's a full MWE.

\documentclass[]{article}
\usepackage{amsmath,amssymb}
\newcommand{\gap}{} %initialise \gap to exist
\newcommand{\GAP}{1}{
    \renewcommand{\gap}{\hspace{##1em}} }
\begin{document}
\GAP{3}
\end{document}

Answer 1

There were three things wrong

1. The syntax for \newcommand requires the number of arguments in square brackets: \newcommand{\GAP}[1]{...}, not curly brackets \newcommand{\GAP}{1}{...}

2. The definition of \gap, since it does not take its own argument, must be specified as #1 (i.e., evaluated within the execution of \GAP), rather than as ##1 (which would imply that it is an argument to \gap)

3. Some stray spaces were in the definition of \GAP.

The MWE:

\documentclass[]{article}
\usepackage{amsmath,amssymb}
\newcommand{\gap}{} %initialise \gap to exist
\newcommand{\GAP}[1]{\renewcommand{\gap}{\hspace{#1em}}}
\begin{document}
a\gap b

\GAP{3}
a\gap b
\end{document}

Answer 2

The number of arguments is an optional argument (in brackets) to \newcommand. Here is the corrected example:

\documentclass[]{article}
\usepackage{amsmath,amssymb}
\newcommand{\gap}{} %initialise \gap to exist
\newcommand{\GAP}[1]{%
    \renewcommand{\gap}{\hspace{#1em}} }
\begin{document}
Text\gap Text\GAP{3}
Text\gap Text
\end{document}