Consider the Kuhn simplicial subdivision for the three-dimensional unit simplex with $D=2$. There are ten vertices of the form
$$
\left(\pi_1 / 2, \pi_2 / 2, \pi_3 / 2, \pi_4 / 2\right) \quad\left(\sum_{i=1}^4 \pi_i=2\right)
$$
to which are attached the following labels:
$$
\begin{array}{ccc}
\text { Vertex } & \text { Coordinates } & \text { Label } \\
\hline \text { A } & (2,0,0,0) & 1 \\
\text { B } & (0,2,0,0) & 2 \\
\text { C } & (0,0,2,0) & 3 \\
\text { D } & (0,0,0,2) & 4 \\
\text { E } & (1,1,0,0) & 2 \\
\text { F } & (0,1,1,0) & 2 \\
\text { G } & (0,0,1,1) & 4 \\
\text { H } & (1,0,1,0) & 3 \\
\text { I } & (0,1,0,1) & 4 \\
\text { J } & (1,0,0,1) & 1
\end{array}
$$
(a) Illustrate this simplex as a pyramid in $\mathbf{R}^3$ with vertices $A, B, C$ as the base and $D$ as the top. Indicate on the diagram the positions of the remaining vertices.
(b) Verify that this labeling satisfies the labeling rule of Sperner's Lemma.
(c) Starting with the base vertex
$$
A=\left(\begin{array}{l}
2 \\
0 \\
0 \\
0
\end{array}\right)
$$
and each possible permutation $\psi:\{1,2,3\} \rightarrow\{1,2,3\}$, find all of the subsimplices contained in the main simplex which have $A$ as a base.
(d) Repeat the procedure of part (c) for each of the remaining vertices $B, C, D, E, F, G, H, I$, and $J$ (as in part (c), you need only determine the subsimplices contained in the main simplex).
(e) List the subsimplices determined in parts (c) and (d), and illustrate the simplicial subdivision in your diagram.
(f) Starting with the base vertex
$$
\left(\begin{array}{l}
2 \\
0 \\
0 \\
0
\end{array}\right)
$$
compute the path to a completely labeled subsimplex. Illustrate the path in your diagram.