Cake number¶

In mathematics, the cake number, denoted by C n , is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly n planes.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8311
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Combinatorial Geometry → Mathematics

Core Idea¶

Cake number is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In mathematics, the cake number, denoted by C n , is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly n planes. In mathematics, the cake number, denoted by C n , is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly n planes. The cake number is so called because one may imagine each partition of the cube by a plane as a slice made.

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Most Pieces of Box Cake

Imagine a cake shaped like a box. You slice it with a big flat knife, going straight through, a few times. The cake number tells you the most pieces you can possibly get with that many slices, if you aim each slice cleverly.

Max Pieces from Flat Cuts

The cake number answers a slicing puzzle. You have a cube-shaped cake and you make a certain number of perfectly flat, straight cuts right through it. The cuts can go in any direction, and you don't rearrange the pieces between cuts. The cake number for n cuts is the largest number of pieces you can possibly end up with. It's the 3D version of a 2D puzzle about cutting a flat pancake with straight lines, called the lazy caterer's sequence.

Maximal Regions of a Cube by Planes

The cake number C_n is the maximum number of regions into which a three-dimensional cube can be divided by exactly n planes. The name comes from picturing each plane as a straight knife cut through a cube-shaped cake. The key word is maximum: the planes have to be arranged in the best way, so that each new cut passes through as many existing pieces as possible. It is the 3D version of the lazy caterer's sequence, which counts the most pieces a flat disk can be cut into with n straight lines. The cake numbers can be written with a formula that uses binomial coefficients in n.

The cake number C_n is the maximum number of regions into which a three-dimensional cube can be partitioned by exactly n planes. It is named for the image of each plane as a single straight knife cut through a cube-shaped cake, and it is the three-dimensional analogue of the lazy caterer's sequence, which gives the maximum number of regions formed by n lines in the plane. The count is extremal: it concerns the best arrangement of the planes, not a typical one, and it counts pieces from exactly n planar cuts without rearranging pieces between cuts. The sequence has a closed form expressible in binomial coefficients of n. A genuine instance of the concept has to keep the cube, the planar cuts, the fixed number n, and the maximization over arrangements.

Scope of Application¶

General formula. If n! denotes the factorial, and we denote the binomial coefficients by.
General formula. and we assume that n planes are available to partition the cube, then the n-th cake number is.
General formula. Cn = {n \choose 3} + {n \choose 2} + {n \choose 1} + {n \choose 0} = \tfrac{1}{6}!\left(n^3 + 5n + 6\right) = \tfrac{1}{6}(n+1)\left(n(n-1) +.
Properties. The cake numbers are the 3-dimensional analogue of the 2-dimensional lazy caterer's sequence.
Properties. The difference between successive cake numbers also gives the lazy caterer's sequence.

Clarity¶

A clear use of Cake number names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, the cake number, denoted by C n , is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly n planes.

Manages Complexity¶

Cake number compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—in mathematics, the cake number, denoted by C n , is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly n planes.—and the practical consequence—cn = {n \choose 3} + {n \choose 2} + {n \choose 1} + {n \choose 0} = \tfrac{1}{6}!\left(n^3 +.

Abstract Reasoning¶

Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematics, the cake number, denoted by C n , is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly n planes.
Check operation and conditions. The values of C n for are given by .
Demand recognition evidence. If n! denotes the factorial, and we denote the binomial coefficients by.
Test variation.

Knowledge Transfer¶

Within the home domain. Knowledge about Cake number transfers literally when a new case preserves the same carrier type, relation, and recognition test. If n! denotes the factorial, and we denote the binomial coefficients by. and we assume that n planes are available to partition the cube, then the n-th cake number is. Beyond the home domain. No canonical parent is asserted for Cake number. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Neighborhood in Abstraction Space¶

Cake number sits in a sparse region of the domain-specific corpus (80^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Geometric Figures & Constructions (32 abstractions)

Nearest neighbors

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