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

TL;DR

The paper reframes the classic egg drop problem to teach recursive problem-solving across dimensions, deriving rigorous worst-case bounds for 1D, 2D, and 3D generalizations: $P_1(k)\le \lceil k N^{1/k}\rceil$, $P_2(k)\le \lceil (k-1)(M+N)^{1/(k-1)}\rceil$, and $P_3(k)\le \lceil (k-2)(L+M+N)^{1/(k-2)}\rceil$, alongside two methods for the 2D critical line problem $x+y=V$ with bounds $L_2^{(1)}(k)\le \lceil k (M+N)^{1/k}\rceil$ and $L_2^{(2)}(k)\le \lceil (k-1) M^{1/(k-1)}\rceil + 1$. A core technical contribution is a general minimizing lemma used to optimize step sizes, enabling clean inductive proofs across dimensions. The work also outlines future directions, including extensions to generic linear forms $\alpha x+\beta y=V$, average-case analyses, and deeper comparisons between competing strategies. Overall, it provides a structured framework of recurrence-based methods to explore high-dimensional variants of a classic combinatorial puzzle and demonstrates how such generalizations can motivate classroom-based research.

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, $αx+βy=V$ (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.

Figures (14)

• Figure 1: Solution to the recursive tiling problem.
• Figure 2: The 1D critical point strategy.
• Figure 3: The 1D critical point strategy in the worst case.
• Figure 4: The 2D critical point strategy.
• Figure 5: The 2D critical point strategy in the worst case.

Theorems & Definitions (11)

• Remark 2.1
• Lemma 3.1
• proof
• Theorem 4.1
• Theorem 5.1
• Conjecture 1
• Theorem 6.1
• Theorem 6.2
• Theorem 6.3
• Theorem B.1