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SummaryCatanis a dynamic property-building and trading board game in which players build a new board every time they play by arranging tiles, number tokens, and port markers. In the interest of creating an equitable board, the official Catanrules restrict how the number tokens are placed. In this paper, we use three techniques to count the number of nonequivalent boards satisfying this restriction, as well as determine the probability that a randomly generated board will be legal.

SummaryConsider the problem of determining, without using a computer or calculator, whether a given quadratic form factors into the product of two linear forms. A solution derived by inspection is often highly nontrivial; however, we can take advantage of equivalent conditions. In this article, we prove the equivalence of five such conditions. Furthermore, we discuss vocabulary such as “reducible,” “degenerate,” and “singular” that are used in the literature to describe these conditions, highlighting the inconsistency with which this vocabulary is applied.