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\begin{center}
{\Large NMU Math \& CS Department} \\
\vspace{0.5cm}
{\Large Problem of the Month, December/January 2024-25}
\end{center}

\vspace{1cm}

\paragraph{} Here we have a (I wouldn't say ``easy'', but quite approachable) christmas-themed problem of the month to think about while school is out for the holiday season! See next page for rules and information.

\vspace{1cm}

\paragraph{} Suppose you are making a circular bracelet using ten beads, where each bead corresponds to one of the letters in the word ``SANTAKLAUS''. Notice that you are free to rotate and turn over the bracelet. Some examples are illustrated below.

\begin{equation*}
\begin{matrix}
\begin{tikzpicture}[scale=1.5]
\node (a) at   (0:1) {S};
\node (b) at  (36:1) {N} edge [-] (a);
\node (c) at (72:1) {A} edge [-] (b);
\node (d) at   (108:1) {K} edge [-] (c);
\node (e) at  (144:1) {S} edge [-] (d);
\node (f) at  (180:1) {A} edge [-] (e);
\node (g) at  (216:1) {L} edge [-] (f);
\node (h) at  (252:1) {A} edge [-] (g);
\node (i) at  (288:1) {T} edge [-] (h);
\node (j) at  (324:1) {U} edge [-] (i) edge [-] (a);
\end{tikzpicture}
&
\raisebox{16mm}{$=$}
&
\begin{tikzpicture}[scale=1.5]
\node (a) at   (0:1) {A};
\node (b) at  (36:1) {S} edge [-] (a);
\node (c) at (72:1) {K} edge [-] (b);
\node (d) at   (108:1) {A} edge [-] (c);
\node (e) at  (144:1) {N} edge [-] (d);
\node (f) at  (180:1) {S} edge [-] (e);
\node (g) at  (216:1) {U} edge [-] (f);
\node (h) at  (252:1) {T} edge [-] (g);
\node (i) at  (288:1) {A} edge [-] (h);
\node (j) at  (324:1) {L} edge [-] (i) edge [-] (a);
\end{tikzpicture}
&
\raisebox{16mm}{$\neq$}
&
\begin{tikzpicture}[scale=1.5]
\node (a) at   (0:1) {A};
\node (b) at  (36:1) {U} edge [-] (a);
\node (c) at (72:1) {K} edge [-] (b);
\node (d) at   (108:1) {N} edge [-] (c);
\node (e) at  (144:1) {A} edge [-] (d);
\node (f) at  (180:1) {S} edge [-] (e);
\node (g) at  (216:1) {A} edge [-] (f);
\node (h) at  (252:1) {S} edge [-] (g);
\node (i) at  (288:1) {T} edge [-] (h);
\node (j) at  (324:1) {L} edge [-] (i) edge [-] (a);
\end{tikzpicture}
\end{matrix}
\end{equation*}

\paragraph{} Notice that the first (2) bracelets are equivalent by turning the bracelet over along the vertical line, and the third bracelet is inequivalent to the previous two.

\paragraph{} How many \textbf{distinct} bracelets like this can be made?

\newpage

\paragraph{} The NMU Mathematics and Computer Science Department invites you to participate in the 2024/2025 Problem of the Month contest to have some fun and get a little recognition. There are paper copies of the problems available at the department front desk if you'd like.

\paragraph{} \textbf{Rules:} Anyone is welcome to submit a solution, and all correct solutions will be recognized; but only undergraduates enrolled in coursework at NMU are eligible to win the prize at the end of the year. The first student to submit a correct solution is the winner of the month. The top problem solver for the academic year will receive a fabulous prize, and be recognized at the department year-end celebration. You must write clear and complete solutions to the problems. You must include your name, NMU IN, address, phone, email, and exact date/time of your submission. You may either submit in person at the department front desk (use a staple if there are multiple pages), or email your solution in \texttt{pdf} format to \texttt{darowe@nmu.edu}.



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