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Post-class fixes

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2024-04-08 07:37:59 -07:00
parent f2ba5bf1b3
commit 759e7e05f6
4 changed files with 24 additions and 19 deletions
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7
Advanced/De Bruijn/parts/1 words.tex
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@ -51,7 +51,7 @@ Find the following:
\begin{solution}
In order from $\mathcal{S}_0$ to $\mathcal{S}_6$:
\begin{itemize}
\item 1, 2, 3, 3, 3, 2, 1
\item 1, 2, 3, 4, 3, 2, 1
\item 1, 3, 5, 4, 3, 2, 1
\end{itemize}
\end{solution}
@ -167,9 +167,10 @@ We'll call this the \textit{Fibonacci word} of order $k$.
\problem{}<cword>
Let $C_k$ denote the word over the alphabet $\{\texttt{0}, \texttt{1}\}$ obtained by \par
concatenating the binary representations of the integers $0,~...,~2^k -1$. \par
For example, $C_1 = \texttt{0}$, $C_2 = \texttt{011011}$, and $C_3 = \texttt{011011100101110111}$.
For example, $C_1 = \texttt{01}$, $C_2 = \texttt{011011}$, and $C_3 = \texttt{011011100101110111}$.
\begin{itemize}
\item How many symbols does the word $C_k$ contain?
% Good bonus problem, hard to find a closed-form solution
% \item How many symbols does the word $C_k$ contain?
\item Compute $\mathcal{S}_0$, $\mathcal{S}_1$, $\mathcal{S}_2$, and $\mathcal{S}_3$ for $C_3$.
\item Show that $\mathcal{S}_k(C_k) = 2^k - 1$.
\item Show that $\mathcal{S}_n(C_k) = 2^n$ for $n < k$.