Problems

Draw a Sperner’s coloring for the following \(3\)-dimensional simplex. The point \(F\) is on the face \(ABC\), point \(E\) is on the face \(BCD\), point \(G\) is on the face \(ACD\) and the point \(H\) is on the face \(ABD\).

Four football teams play in a tournament. There’s the Ulams (\(U\)), the Vandermondes (\(V\)), the Wittgensteins (\(W\)) and the Xenos (\(X\)). Each team plays every other team exactly once, and matches can end in a draw.
If a game ends in a draw, then both teams get \(1\) point. Otherwise, the winning team gets \(3\) points and the losing team gets \(0\) points. At the end of the tournament, the teams have the following points totals: \(U\) has \(7\), \(V\) has \(4\), \(W\) has \(3\) and \(X\) has \(2\).

Work out the results of each match, including showing that there’s no other way the results could have played out.