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\title{MATH 3040: Preliminary Exam \#1, Part II}
\author{}
\date{October 4, 2017}

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All answers must be in complete sentences. All assertions must be proven, per the student contract. Remember that this is a writing class, so while correctness and thoroughness are most important, \textit{style} matters as well. 

For this part of the exam, you are free to use any of the allowable resources for the course (e.g. notes, book, your previous homeworks, etc.), but you are \textbf{not allowed to discuss the questions with each other}.

This part of the exam is due on Gradescope by Sunday, October 8.

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\textbf{1. (Generalized Meet-First Strategy)} Recall the Meet-First game from the HW \#2. In groups, you answered the question of who could win if the bridge was $n$ feet long and you could move up to $3$ feet in one turn. What about if the bridge were $n$ feet long and you could move up to $k$ feet in one turn (that is, in a single turn a player could move 1 foot, 2 feet, 3 feet, $\ldots$, $k-1$ feet, or $k$ feet)? Namely, for which pairs $(n,k)$ can either player guarantee a win?


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\textbf{2. (Cash Register)} Show that for any natural number $n \geq 4$, you can get exactly $n$ dollars by using only \$2 bills and \$5 bills.

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