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\flushleft \underline{Homework 10 -- MATH 2510 Spring 2019}\\
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\begin{enumerate}[1.]
\item Recall the definition of the Ackermann function from the notes. Compute $\text{Ack}(2,3)$.
\item Consider a function defined recursively by the formula
\[ \left\{ \begin{array}{ll}
x(0)=2 \\
x(n+1)=3x(n) 
\end{array} \right.\]
What is $x(4)$? What is $x(5)$? In general, what is the (non-recursive) formula for $x(n)$? \\
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Consider the theory of four-point geometry defined by the following axioms:
\begin{tabular}{|ll|}
\hline
\textbf{Axiom 1} & There exist exactly four points. \\
\textbf{Axiom 2} & Each two distinct points have exactly one line that contains both of them. \\
\textbf{Axiom 3} & Each line is exactly on two points. \\
\hline
\end{tabular}
\item Draw a picture (``model") of four-point geometry.
\item Show that the axioms of four-point geometry are all independent from each other. \\
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Consider six-line geometry defined by the following axioms:
\begin{tabular}{|ll|}
\hline
\textbf{Axiom 1} & There exist exactly six lines. \\
\textbf{Axiom 2} & Each two distinct lines have exactly one point on both of them. \\
\textbf{Axiom 3} & Each point is on exactly two lines. \\
\hline
\end{tabular}
\item Draw a model of six-line geometry.
\item Show that the axioms of six-line geometry are all independent from each other. \\
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Consider Fano's geometry defined by the following five axioms:
\begin{tabular}{|ll|}
\hline
\textbf{Axiom 1} & There exists at least one line. \\
\textbf{Axiom 2} & Every line has exactly 3 points on it. \\
\textbf{Axiom 3} & Not all points are on the same line. \\ 
\textbf{Axiom 4} & For two distinct points, there exists exactly one line on both of them. \\
\textbf{Axiom 5} & Each two lines have at least one point on both of them. \\
\hline
\end{tabular}
\item Draw a model for Fano geometry (hint: letting your ``lines" be curved can be helpful!)

\end{enumerate}
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