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Mark
2025-02-26 08:46:48 -08:00
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src/Advanced/Error-Correcting Codes/parts/00 detection.tex
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@@ -1,6 +1,8 @@
\section{Error Detection}
An ISBN\footnote{International Standard Book Number} is a unique numeric book identifier. It comes in two forms: ISBN-10 and ISBN-13. Naturally, ISBN-10s have ten digits, and ISBN-13s have thirteen. The final digit in both versions is a \textit{check digit}.
An ISBN\footnote{International Standard Book Number} is a unique identifier publishers assign to their books. \par
It comes in two forms: ISBN-10 and ISBN-13. Naturally, ISBN-10s have ten digits, and ISBN-13s have thirteen.
The final digit in both versions is a \textit{check digit}.
\vspace{3mm}
@@ -15,7 +17,8 @@ If $n_{10}$ is equal to 10, it is written as \texttt{X}.
\problem{}
Only one of the following ISBNs is valid. Which one is it?
Only one of the following ISBNs is valid. Which one is it? \par
\note[Note]{Dashes are meaningless.}
\begin{itemize}
\item \texttt{0-134-54896-2}
@@ -23,15 +26,16 @@ Only one of the following ISBNs is valid. Which one is it?
\end{itemize}
\begin{solution}
The first has an inconsistent check digit.
The second is valid.
\end{solution}
\vfill
\pagebreak
\problem{}
Take a valid ISBN-10 and change one digit. Is it possible that you get another valid ISBN-10? \par
Provide a proof.
Take a valid ISBN-10 and change one digit. \par
Is it possible that you get another valid ISBN-10? \par
Provide an example or a proof.
\begin{solution}
Let $S$ be the sum $10n_1 + 9n_2 + ... + 2n_9 + n_{10}$, before any digits are changed.
@@ -50,9 +54,8 @@ Provide a proof.
\vfill
\problem{}
Take a valid ISBN-10 and swap two adjacent digits. When will the result be a valid ISBN-10? \par
This is called a \textit{transposition error}.
Take a valid ISBN-10 and swap two adjacent digits. This is called a \textit{transposition error}. \par
When will the result be a valid ISBN-10?
\begin{solution}
Let $n_1n_2...n_{10}$ be a valid ISBN-10. \par
@@ -68,7 +71,8 @@ This is called a \textit{transposition error}.
\pagebreak
\definition{}
ISBN-13 error checking is slightly different. Given a partial ISBN-13 $n_1 n_2 n_3 ... n_{12}$, the final digit is given by
ISBN-13 error checking is slightly different. \par
Given a partial ISBN-13 with digits $n_1 n_2 n_3 ... n_{12}$, the final digit is given by
$$
n_{13} = \Biggr[ \sum_{i=1}^{12} n_i \times (2 + (-1)^i) \Biggl] \text{ mod } 10
@@ -127,7 +131,7 @@ Take a valid ISBN-13 and swap two adjacent digits. When will the result be a val
\vfill
\problem{}<isbn-nocorrect>
\problem{}<isbnnocorrect>
\texttt{978-008-2066-466} was a valid ISBN until I changed a single digit. \par
Can you find the digit I changed? Can you recover the original ISBN?