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Back in high school, I came across the following contest problem:

Question: Let $S$ be a set of positive integers totaling 20.  What is the maximum value of

$\displaystyle\prod_{x \in S} x$

It’s a fun problem, so don’t rush past the spoiler tags too fast.  When you’re ready, I’ll spoil the solution to the question above, and discuss a “continuous” version of the question above.  Namely, what happens when $S$ is allowed to include positive real numbers?

In its early years, the rigorous study of games (game theory) looked only upon games of so-called perfect information, in which each player knows the moves carried out by all players.  Such games laid the foundation for early economic models of perfect competition, wherein consumers have full knowledge of both market conditions and each others’ consumer tendencies.

Of course, perfect competition — taken literally — cannot exist, and the failure of this model and others prompted economic theorists to study a more-general class of games, games of limited information (also known as games of imperfect information).

In this post, we’ll look at one-player games of limited information (sometimes classified as puzzles, not games) through a topological lens, and create for each game a poset of topologies under which topologically indistinguishable points correspond to outcomes that are indiscernible in a limited-information context.  Expanding this dictionary, we’ll describe a topology on the outcome space under which the “safe” or “warranted” extension of one’s limited information relates to the continuity of certain maps.

In 1832, Galois introduced the concept of normal subgroups, and proved that the groups $A_n$ (for $n>4$) and $\mathrm{PSL}_2(\mathbb{F}_q)$ (for $q>4$) were simple, i.e. admit no non-trivial (proper) normal subgroups.  In 1892, Hölder asked for a classification of all finite simple groups, and the final classification of finite simple groups in 2004 by Aschbacher and Smith’s resolution of the quasithin case thus resolves an open problem over a century in the making. (Note: the great majority of the classification theorem concluded in the 1980’s.  A computational error was identified and resolved in 2008.)

One immediate consequence of the classification of finite simple groups (CFSG) is that the set S given by

$S:=\{n \in \mathbb{N} : \text{there exists a simple group of order } n\}$

has natural density 0.  Yet it seems unlikely that this result requires the tens of thousands of pages currently involved in the proof of the CFSG, and for the rest of this article we seek to bound the density of S using arithmetic methods.

For what follows, we define a loaded die as a discrete probability distribution with six outcomes (labelled {1,…,6}), each of which has positive probability. To each such die, we associate a generating polynomial, given by

$p(t):=p_1t+p_2t^2+\ldots +p_6 t^6$

in which $p_i$ denotes the probability of the outcome i.  If $q(t)$ corresponds to another such die, we note that the product

$p(t)q(t) =\sum_i a_i t^i:=\sum_{i=2}^{12}\big( \sum_{k=1}^{i-1} p_k q_{i-k}\big) t^i$

has coefficients which reflect the probabilities of  certain dice sums for p and (and this is the utility of generating polynomials).  We are now ready to ask the following question:

Question: Does there exist a pair of loaded dice such that the probability of rolling any dice sum ({2,…12}) is equally likely?