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\section{Induction II}

\noindent{\bf{Problem 1:}} We've stated in class that any finite set of natural numbers has a least element. Prove this using induction.~\\[10pt]

\noindent{\bf{Problem 2:}} Let the \emph{height} of a binary tree be the number of nodes on the longest path from the root to the leaves. Let the \emph{size} of the tree be the total number of nodes. Prove that the size of a binary tree is always less than $2^\textit{height}$.~\\[10pt]

\noindent{\bf{Problem 3:}} Two people are playing a game of hot potato. They can each hold the potato for $1$, $2$, or $3$ seconds before passing it to their opponent. The potato starts with some $n$ seconds on the clock (which both players know) and the player holding the potato when it hits $0$ loses. (If a player throws it when it hits $0$, the other player loses.) Under which conditions can the first player ensure victory? Prove it.
~\\[10pt]

\noindent{\bf{Problem 4:}} ~\\[5pt]
\noindent(a) Use the principle of weak induction to prove the principle of strong induction.~\\[10pt]
\noindent(b) Use the principle of strong induction to prove the principle of weak induction.~\\[10pt]

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