Egg Drop Problems: They Are All They Are Cracked Up To Be!

TL;DR

This paper explores higher-dimensional generalizations of the classical egg drop problem, providing new bounds for the minimal number of trials in multiple dimensions and analyzing related critical line problems to enhance educational engagement.

Contribution

It introduces recursive algorithms and bounds for multi-dimensional egg drop problems, extending classical results and proposing a conjecture for the general case, along with analysis of critical line problems.

Findings

Derived bounds for 2D and 3D egg drop problems

Proposed a conjecture for the general d-dimensional case

Analyzed critical line problems and solution strategies

Abstract

We illustrate how to invite and excite students about research by exploring higher-dimensional generalizations of the classical egg drop problem, in which the goal is to locate a critical breaking point using the fewest number of trials. Beginning with the one-dimensional case, we prove that with $k$ eggs and $N$ floors, the minimal number of drops in the worst case satisfies $P_1(k) \leq \lceil k N^{1/k} \rceil$. We then extend the recursive algorithm to two and three dimensions, proving similar formulas: $P_2(k) \leq \lceil (k-1)(M+N)^{1/(k-1)} \rceil $ in 2D and $P_3(k) \leq \lceil (k-2)(L+M+N)^{1/(k-2)} \rceil$ in 3D, and conjecture a general formula for the $d$-dimensional case. Beyond the critical point problems, we then study the critical line problems, where the breaking condition occurs along $x+y=V$ (with slope $-1$) or, more generally, $\alpha x+\beta y=V$ (with the slope of the line unknown). We discuss how one frequently has to pivot from the original problem, which is intractable, to something that can be solved; in our case, using induction and recursion, two standard proof techniques.