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I've tried pasting "圍棋 (wéiqí)" into my editor, but after typesetting I see only "?? (wiq)" in the PDF. What do I need to change to fix these characters?

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You need to use an TeX engine (XeTeX or LuaTeX) that handles Unicode input correctly and a font that contains the characters. You also need to make sure that your file is saved as UTF-8.

See the following questions for help on these issues:

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You can try the {CJKutf8} package according to this blog.

\documentclass{article}

\usepackage{CJKutf8}

\begin{document}
\begin{CJK}{UTF8}{gbsn}
围棋 (wéiqí)
\end{CJK}
\end{document}

Note that, because of the fonts, the traditional Chinese character "圍" is not correctly displayed. So I am using the simplified Chinese character "围" here.

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