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 mathematics_logic_statistics 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 by a knife through a cube-shaped cake. It is the 3D analogue of the lazy caterer's sequence.

The values of C n for are given by . 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.

For Cake number, the abstraction is narrower than the article's general subject matter: a positive case must preserve 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. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

How would you explain it like I'm…

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.

Structural Signature¶

Sig role-phrases:

Defining carrier — The cake number is so called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake.
Constitutive relation — 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.
Operating condition — The values of C n for are given by .
Recognition evidence — If n! denotes the factorial, and we denote the binomial coefficients by.
Admissible variation — and we assume that n planes are available to partition the cube, then the n-th cake number is.
Characteristic consequence — C_n = {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) + 6\right).
Failure boundary — The cake numbers are the 3-dimensional analogue of the 2-dimensional lazy caterer's sequence.

What It Is Not¶

Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by 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.
Not an over-broad reading. In n spatial (not spacetime) dimensions, Maxwell's equations represent C_n different independent real-valued equations.
Not an over-broad reading. If n! denotes the factorial, and we denote the binomial coefficients by.
Not an over-broad reading. and we assume that n planes are available to partition the cube, then the n-th cake number is.
Not automatically Taxicab number. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Cake number applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

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. C_n = {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) + 6\right).
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.
Properties. The fourth column of Bernoulli's triangle (k = 3) gives the cake numbers for n cuts, where n ≥ 3.

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

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. The strongest recognition evidence in the frozen account is: If n! denotes the factorial, and we denote the binomial coefficients by. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification In n spatial (not spacetime) dimensions, Maxwell's equations represent C_n different independent real-valued equations. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Cake number compresses multiple mathematics_logic_statistics 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—c_n = {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) + 6\right). This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics_logic_statistics 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. Change an implementation or setting while preserving and we assume that n planes are available to partition the cube, then the n-th cake number is.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

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.

Examples¶

Canonical¶

If n! denotes the factorial, and we denote the binomial coefficients by. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → 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; recognition evidence → If n! denotes the factorial, and we denote the binomial coefficients by

Applied / In Practice¶

and we assume that n planes are available to partition the cube, then the n-th cake number is. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → General formula; invariant → 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; boundary → the case exits the class when in n spatial (not spacetime) dimensions, Maxwell's equations represent C_n different independent real-valued equations

Structural Tensions¶

T1 — Stable identity versus admissible variation. In n spatial (not spacetime) dimensions, Maxwell's equations represent C_n different independent real-valued equations. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. If n! denotes the factorial, and we denote the binomial coefficients by. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. and we assume that n planes are available to partition the cube, then the n-th cake number is. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. C_n = {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) + 6\right). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. The cake number is so called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Cake number literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. 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 tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Cake number distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Cake number is structural-leaning. Its structural side is the repeatable organization summarized by 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. Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: The values of C n for are given by . Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. 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 stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: The cake number is so called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake. 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. It further constrains recognition and variation through: The values of C n for are given by . If n! denotes the factorial, and we denote the binomial coefficients by.

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Cake number literal. Its documented scope includes the condition that If n! denotes the factorial, and we denote the binomial coefficients by. Another bounded application condition is that and we assume that n planes are available to partition the cube, then the n-th cake number is. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—and we assume that n planes are available to partition the cube, then the n-th cake number is.—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Cake number. The reviewed identity 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. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

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

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish 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?
Taxicab number. The smallest positive integer expressible as a sum of two positive cubes in a specified number of distinct unordered ways. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Diophantine Equation. A polynomial equation with integer coefficients whose admissible solutions are required to be integers. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Arithmetic number. A positive integer whose positive divisors have an integer arithmetic mean. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Cake number remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Cake_number (revision 1339859439).

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.