Disk-Covering Problem¶

One of the covering disks is placed central and the remaining five in a symmetrical way around it.

Version: v1 · 2026-09-28 · History
Domain-specific #: 9010
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Discrete Geometry, Covering Problems → Mathematics

Core Idea¶

Disk-Covering Problem is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: One of the covering disks is placed central and the remaining five in a symmetrical way around it. The disk covering problem asks for the smallest real number r(n) such that n disks of radius r(n) can be arranged in such a way as to cover the unit disk. Dually, for a given radius ε, one wishes to find the smallest integer n such that n disks of radius ε can cover the unit disk.

How would you explain it like I'm…

The Sticker-Cover Puzzle

Imagine a round cookie that you want to cover completely with smaller round stickers, all the same size. If you get a certain number of stickers, how small can they be and still cover the whole cookie? For some numbers of stickers, the best way people have found is to put one sticker in the middle and the rest in a neat ring around it.

Smallest Circles to Cover a Circle

The disk-covering problem is a math puzzle about circles. You have one circle of size 1, and a set number of smaller circles that are all the same size. The question is: what's the smallest size those circles can be and still cover every bit of the big circle? You can also ask it the other way: if the small circles have a fixed size, what's the fewest you need? For seven to ten circles, the best arrangements known put one circle in the center with the others spaced evenly around it.

Minimal-Radius Disk Covering

The disk-covering problem asks for the smallest radius r(n) such that n equal disks of that radius can be arranged to cover the unit disk completely. The dual version fixes the radius ε and asks for the smallest number of disks of radius ε that can cover the unit disk. Solutions are given as the best known arrangements for each n. One notable pattern places one disk at the center and the rest symmetrically around it: one central disk with five around it is not the best layout for r(6), but one central disk with six, seven, eight, or nine disks around it gives the best known layouts for r(7), r(8), r(9), and r(10). The positions in such arrangements are described by the angles of the surrounding disks.

The disk-covering problem seeks, for each n, the smallest real number r(n) such that n disks of radius r(n) can be placed to cover the unit disk; dually, for a given radius ε, it seeks the minimal number n of radius-ε disks that cover the unit disk. Results are reported as best-known solutions, specified by the disk arrangement and the resulting radius. A recurring structure is a central disk surrounded by a symmetric ring of equal disks, with the ring's angular positions recorded as a symmetry parameter. For six disks, a central disk plus five symmetrically placed disks is not the optimal layout. However, analogous arrangements with six, seven, eight, and nine disks around a central disk of the same radius provide the best known layouts for r(7), r(8), r(9), and r(10). The problem is a specific covering question about equal disks and the unit disk, not covering problems in general.

Scope of Application¶

Method. The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6.
Method. One of the covering disks is placed central and the remaining five in a symmetrical way around it.
Method. The corresponding angles θ are written in the "Symmetry" column in the above table.
Method. While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best.
Documented setting. The disk covering problem asks for the smallest real number r(n) such that n disks of radius r(n) can be arranged in such a way as to cover the.

Clarity¶

A clear use of Disk-Covering Problem names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is One of the covering disks is placed central and the remaining five in a symmetrical way around it.

Manages Complexity¶

Disk-Covering Problem compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—one of the covering disks is placed central and the remaining five in a symmetrical way around it.—and the practical consequence—dually, for a given radius ε, one wishes to find the smallest integer n such that n disks of radius ε can cover the unit disk.

Abstract Reasoning¶

Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: One of the covering disks is placed central and the remaining five in a symmetrical way around it.
Check operation and conditions. The corresponding angles θ are written in the "Symmetry" column in the above table.
Demand recognition evidence.

Knowledge Transfer¶

Within the home domain. Knowledge about Disk-Covering Problem transfers literally when a new case preserves the same carrier type, relation, and recognition test. The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6. One of the covering disks is placed central and the remaining five in a symmetrical way around it. Beyond the home domain. No canonical parent is asserted for Disk-Covering Problem.

Neighborhood in Abstraction Space¶

Disk-Covering Problem sits in a moderately populated region (57^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Combinatorial Optimization & Discrete Structures (31 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08