Limit theory of discrete mathematics problems

Abstract

We show a general problem-solving tool called limit theory. This is an advanced version of asymptotic analysis of discrete problems when some finite parameter tends to infinity. We will apply it on three closely related problems.Alpern's Caching Game (for 2 nuts) is defined as follows. The hider caches 2 nuts into one or two of n potential holes by digging at most 1 depth in total. The goal of the searcher is to find both nuts in a limited time h, otherwise the hider wins. We will show that if h and n/h are large enough, then very counterintuitively, any optimal hiding strategy should dig less than 1 in total, with positive probability. We will prove it by defining and analyzing a limit problem. Then we will partially solve the entire problem. We will also have significant progress with two other problems: the Manickam-Miklós-Singhi Conjecture and the Kikuta-Ruckle Conjecture.

Optimizing stakes in simultaneous bets

“…An array of triangles along the hypothenuse remains to be filled in that part. Numerical results of Csóka (2015) suggest that bold play is optimal for all of these triangles, except for the one touching on {(p, p) : 1 2 ≤ p ≤ 2 3 }. In the next section we will confirm that bold play is not optimal for this particular triangle.…”

“…Conjecture 1.1 (Csóka (2015)). For every p and t there exists a k ∈ N such that π(p, t) is realized by c i = 1 k if i ≤ k and c i = 0 if i > k for some k ∈ N. In other words, the maximal probability is realized by an average.…”