Assignment:
Find a simplex, a simplicial partition of this simplex, and a nonproper coloring of the simplicial partition, for which the number of perfectly colored n-dimensional simplices is odd. In other words, show that in Sperner’s Lemma the properness of the coloring is a sufficient but not necessary condition for this number to be odd.
Provide complete and step by step solution for the question and show calculations and use formulas.