diff --git a/latex/appendices.tex b/latex/appendices.tex
index e1f55de..3525403 100644
--- a/latex/appendices.tex
+++ b/latex/appendices.tex
@@ -3,11 +3,41 @@
 \begin{document}
 \appendix
 \section{Ising model system states}\label{sec:system_states}
+To avoid counting duplicates, we used 
+\begin{figure}\label{fig:tikz_counting}
+    \centering
+    \begin{tikzpicture}
+        \draw (0, 0) grid (2, 2);
+        % \node[inner] (s1) at (0.5, 0.5) {s};
+        \node (s1) at (0.5, 1.5) {$s_1$};
+        \node (s2) at (1.5, 1.5) {$s_2$};
+        \node (s3) at (0.5, 0.5) {$s_3$};
+        \node (s4) at (1.5, 0.5) {$s_4$};
+        \node[gray] (s12) at (2.5, 1.5) {$s_1$};
+        \node[gray] (s34) at (2.5, 0.5) {$s_3$};
+        \node[gray] (s13) at (0.5, -0.5) {$s_1$};
+        \node[gray] (s24) at (1.5, -0.5) {$s_2$};
+    
+        \draw[red, ->] (s1.east) -- (s2.west);
+        \draw[red, ->] (s2.east) -- (s12.west);
+        \draw[red, ->] (s1.south) -- (s3.north);
+        \draw[red, ->] (s2.south) -- (s4.north);
+        \draw[red, ->] (s3.east) -- (s4.west);
+        \draw[red, ->] (s4.east) -- (s34.west);
+        \draw[red, ->] (s3.south) -- (s13.north);
+        \draw[red, ->] (s4.south) -- (s24.north);
+      \end{tikzpicture}
+      \caption{Rules for multiplying spin pairs.}
+\end{figure}
 
 
 \section{Analytical expressions}\label{sec:analytical_expressions}
-
-
+The Boltzmann distribution is normalized using a partition function $Z$ given by
+\begin{align*}
+	Z &= \sum_{\text{all possible } s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\
+	&= \dots \\
+	&= 4 \cosh (8 \beta J) + 12 \\
+\end{align*}
 
 
 \end{document}
\ No newline at end of file