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Greedy Packing of Nested Rings: Placement Rules, a Golden Counterexample, and a Tribonacci Floor

The greedy approach returns the lexicographically maximal feasible set for planar disks when rho <= phi; in the independent-holes model, 1/sqrt(2) is the sharp threshold for area optimality. This golden guarantee holds for any finite inventory of planar disks, while in higher dimensions, the guarantee is limited to packing at most five rings.

Original authors: Javier Aguilar Martín

Published 2026-09-15✓ Author reviewed ⓘ

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Javier Aguilar Martín

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a kitchen where you are frying squid rings. You have a large pan and a pile of rings of various sizes. Some rings are wide and flat; others are narrow and small. The goal is to fit as many rings as possible into the pan without them overlapping. There is a clever trick: a small ring can sit perfectly inside the hollow center of a larger ring, nesting like a set of Russian dolls. This simple physical setup creates a complex puzzle for mathematicians. They want to know if a simple, step-by-step strategy works best. The strategy is to take the rings one by one, starting with the largest, and place each one wherever it fits. If a ring can fit inside the hole of a larger ring already in the pan, you put it there; otherwise, you place it on the empty floor of the pan. The question is whether this greedy approach always leads to the best possible result, or if a smarter, more complicated plan is needed to pack more rings or to maximize the total surface area touching the pan.

This puzzle belongs to a field of mathematics called geometry, specifically the study of how shapes fit together in space. For decades, mathematicians have known that for certain types of packing problems, a simple greedy rule works perfectly. However, when the shapes are rings that can nest inside one another, the rules change. The new research shows that the answer depends entirely on how the sizes of the rings relate to each other. If the rings are sized in a very specific way—where the sum of the radii of all smaller rings is very small compared to the current ring—the simple greedy strategy is guaranteed to be perfect. In this scenario, while you must still process the rings from largest to smallest, you have total freedom in where you place each ring; no matter which available hole or spot you choose, you will always end up with the same set of rings, the lexicographically maximal one.

However, the researchers discovered that this perfect behavior has a sharp limit. When the rings are not quite so drastically different in size, the simple greedy strategy can fail. They proved that if you have four rings, the greedy method might miss the optimal solution, even if the rings are sized in a way that seems almost safe. The point where the strategy stops working is tied to a famous number known as the golden ratio, approximately 1.618. The study shows that as long as the ratio of the sum of the smaller radii to the current radius is less than or equal to this golden number, the greedy method is safe for returning the lexicographically maximal set. But if the smaller rings get larger relative to the current ring, the simple strategy can break down, leaving rings on the table that could have been packed.

The team also found that this failure is not just a fluke of a specific arrangement. They constructed pairs of nearly identical situations where the only difference is the size of the smallest rings, yet the greedy method makes the wrong choice in one case and the right choice in the other. Because the algorithm cannot tell these two situations apart just by looking at the current state of the pan, no simple rule based on immediate observation can ever be perfect for all cases. The researchers also explored what happens if the rings have different thicknesses or if the container is a square instead of a circle. They found that while the golden ratio remains a critical threshold for circular pans, for square pans, the threshold is known to be no higher than approximately 1.6845, though the exact number is still being investigated.

Ultimately, the work provides a clear map of when a simple, intuitive approach works and when it fails. It confirms that for a wide range of sizes, the greedy method is not just a good guess but a mathematically proven optimum for superadditive objectives such as contact area. It also pinpoints exactly where that certainty ends, revealing a boundary defined by the golden ratio. This result is significant because it moves beyond computer simulations to provide rigorous, written proofs. While the golden-ratio guarantee holds for any finite inventory of planar disks, the researchers also provide proofs for higher-dimensional spheres, though the golden ratio guarantee in those higher dimensions is currently proven for up to five rings. The study settles a long-standing question about the reliability of greedy packing, showing that while simplicity often wins, there is a precise, beautiful mathematical line where complexity takes over.

Technical Summary: Greedy Packing of Nested Rings

Problem Statement
The paper investigates the problem of packing annuli (rings) of a common width $w$ into a compact container (the "pan"), allowing rings to nest inside the holes of larger rings. The study focuses on two competing objectives: maximizing the number of placed rings (cardinality) and maximizing the total contact area between rings and the container. These objectives diverge because contact area is superadditive in the radius, whereas cardinality is not.

The central question is to determine the conditions under which a descending greedy algorithm (processing rings from largest to smallest radius) is guaranteed to find the lexicographically maximal feasible set of rings, regardless of the specific rule used to choose among feasible containers (e.g., best-fit, worst-fit, or random). This property is termed placement obliviousness. The paper seeks to identify the exact threshold $\rho$ (the maximum ratio of the sum of smaller radii to a given radius) below which this obliviousness holds, and to characterize the geometric and combinatorial structures that cause greedy algorithms to fail.

Methodology
The author employs a hybrid approach combining rigorous geometric analysis, combinatorial optimization, and formal verification:

Geometric Analysis: The paper utilizes properties of disk packing, including Descartes' circle theorem, angular separation criteria for wall-tangent circles, and "row" constructions (packing balls along a diameter). Key tools include the "row lemma" (Lemma 3) and specific "pocket" constructions (Descartes pockets) that define the limits of packing three or more disks.
Combinatorial Reduction: The problem is decomposed into a geometric layer (feasibility of sibling packings) and a combinatorial layer (selection of rings). The author proves that under superincreasing radii (where each radius exceeds the sum of all smaller ones), the combinatorial layer becomes trivial, reducing the problem to a selection greedy that is optimal for any superadditive objective.
Exchange Arguments: To prove placement obliviousness, the author uses a "parent alignment" technique. They demonstrate that if a greedy execution differs from an optimal witness, one can construct a new witness by exchanging the placement of the largest disagreeing ring and re-inserting displaced smaller rings into the vacated space, preserving feasibility.
Formal Verification: The paper relies heavily on computational verification. A repository contains 122 Lean theorems formalizing algebraic identities, polynomial inequalities, and specific geometric certificates. Numerical checks are used to explore parameter spaces and generate counterexamples, but the author explicitly states that numerical results do not substitute for written proofs or formal certificates.

Key Contributions and Results

Superadditivity Dichotomy (Theorem 6): Under superincreasing radii, the selection greedy algorithm is optimal for any positive, strictly increasing, superadditive objective function (including contact area). However, for cardinality (which is not superadditive), counterexamples exist even with superincreasing radii.
Placement Obliviousness under Superincreasing Radii (Theorem 9): If radii are superincreasing, any rule for choosing among feasible containers yields the lexicographically maximal feasible set. This holds for arbitrary container shapes and dimensions.
Sharpness at $n=4$ (Theorem 13): Placement obliviousness holds unconditionally for up to three rings but fails for four rings. The paper constructs "twin instances" (Theorem 14) where the observable state (container capacities, occupants, incoming ring) is identical for two different problem instances, yet the optimal placement for the critical ring differs. This proves that no deterministic state-based rule can succeed universally, and randomized rules fail with probability at least $1/2$ .
The Golden Threshold ( $\tau = \phi$ ):
- Additive Model: The threshold for the additive surrogate model is exactly $\rho = 1$ .
- Geometric Model (Disks): The paper proves that the exact global threshold for disk containers is the Golden Ratio $\phi \approx 1.618$ .
- Counterexamples: A "golden family" of four-ring instances is constructed where placement obliviousness fails at $\rho = \phi + 3\varepsilon$ . This refutes the natural conjecture that the threshold is the Tribonacci constant $T \approx 1.839$ .
- Universal Guarantee (Theorem 22): For any finite inventory of planar disks with $\rho \le \phi$ , every descending greedy execution returns the lexicographically maximal set.
The Tribonacci Floor: While the global threshold is $\phi$ , the paper identifies a "rigid subfamily" of four-ring counterexamples where the infimum of $\rho$ is exactly the Tribonacci constant $T$ . This represents the exact floor for a specific class of blocked exchanges involving rigid geometric configurations.
Square Pans: For square containers, the paper establishes an upper bound for the threshold $\tau_{\square} \le Y \approx 1.6845$ , derived from a specific algebraic family. The exact global threshold for squares remains an open problem.
Dimensionality: Results for spherical containers in dimensions $d \ge 2$ are transferred from the planar case via a dimension-reduction lemma. Placement obliviousness holds for up to three rings in any dimension and fails at four. A separate argument extends the golden threshold guarantee to up to five rings in any dimension.
Independent Holes and Area: For independent hole radii, the paper proves an exact area guarantee factor of $\min(1, \kappa^{-2} - 1)$ under $\rho \le \kappa < 1$ , with a sharp optimality threshold at $1/\sqrt{2}$ .

Significance and Claims
The paper claims to resolve the structural questions regarding greedy packing of nested rings, moving beyond heuristic approaches to provide exact thresholds and impossibility results.

Optimality of Greedy: It establishes that the descending greedy algorithm is placement oblivious (guaranteeing the lexicographically maximal set) for planar disks when $\rho \le \phi$ . Area optimality is a separate property that holds under specific conditions of common width and superincreasing radii.
Geometric vs. Combinatorial: The work decouples the hardness of the problem into a geometric layer (NP-hard in general) and a combinatorial layer (solved by superincreasing conditions).
Refutation of Conjectures: It explicitly refutes the conjecture that the Tribonacci constant is the universal threshold, showing instead that the Golden Ratio is the correct bound for the general case, while Tribonacci remains the floor for a specific rigid subfamily.
Rigorous Verification: The author emphasizes that their results are backed by a combination of written geometric proofs and formalized algebraic certificates in Lean, distinguishing their work from purely numerical or heuristic studies in the circle packing literature.

The paper concludes that while the global threshold for disks is settled at $\phi$ , open problems remain regarding the exact threshold for square pans, the precise lower floors of specialized nested-exchange families beyond the rigid case, and the computational complexity of finding optimal placements when full input access is allowed.

Technical Summary: Greedy Packing of Nested Rings

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