cake number
Let an agent explain cake numbers and their formula, relay the derivation and related sequences from mathematical references, describe puzzle applications, and distinguish cake numbers from the lazy caterer sequence and other cutting problems.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Understand What a cake number is.
Definition
Definition.
Definition
Definition.
- What is a cake number, and what are the first terms of the sequence? definition
- Is the question about cake numbers, the lazy caterer sequence or another cutting problem? boundary
Formula
Formula.
Formula
Formula.
- How is the formula (n^3 + 5n + 6) / 6 derived, and why does it equal a sum of binomial coefficients? action
- Which entry fits binomial coefficient? action
Explore Related problems.
Two dimensions
Lazy caterer.
Two dimensions
Two dimensions.
- How does the two-dimensional lazy caterer sequence relate, and how do the formulas generalise to higher dimensions? provenance
- Which references are standard? provenance
Constraints
Constrained cuts.
Constraints
Constraints.
- What happens if cuts must pass through a point or produce equal pieces? provenance
- Which sources are cited? provenance
Apply Puzzles and teaching.
Puzzles
Puzzles.
Puzzles
Puzzles.
- How do cake-cutting puzzles use cake numbers, and what are common wrong answers? action
- Which entry fits recreational mathematics? action
Teaching
Teaching.
Teaching
Teaching.
- How can cake numbers be used to teach recurrences and combinatorics? provenance
- Which entry fits mathematics education? action
Context History.
History
History.
History
History.
- Where did the problem originate, and who studied hyperplane arrangements? provenance
- Which entry fits hyperplane arrangement? action
Computation
Computation.
Computation
Computation.
- How can cake numbers be computed and verified programmatically? action
- Which entry fits integer sequence? action
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
What it is Filled
The maximum number of regions into which a three-dimensional cube or cake can be divided by n planar cuts, given by the formula (n^3 + 5n + 6) / 6, producing the sequence 1, 2, 4, 8, 15, 26, 42, 64 and so on, the three-dimensional analogue of the lazy caterer sequence for cutting a circle with lines; cake numbers are a classic example in combinatorial geometry and appear in puzzles about cutting a cake into the most pieces.
Why it exists Filled
Let an agent explain cake numbers and their formula, relay the derivation and related sequences from mathematical references, describe puzzle applications, and distinguish cake numbers from the lazy caterer sequence and other cutting problems.
Distinguishing features Filled
- Cubic growth
- Binomial sum form
- Three-dimensional cutting
- Puzzle origin
What robots and AI may and may not do Filled
Must not
- Confuse the cake number with the lazy caterer's number for the plane.
- Present wrong terms or a wrong formula as established.
- Treat the name as describing actual cake cutting or portion sizes.
Only with a human decision
- Publishing corrections to shared reference entries for the sequence.
May
- Compute cake numbers from the formula and check them against the known sequence.
- Use the cake number problem to teach counting regions and dimensions.
Moral aspects Filled
- Teaching material should be accurate because students rely on it.
- Credit for the result belongs to those who established it.
Who is affected
- Students and teachers
- Puzzle readers
Owners Filled
Steward
Nobody owns the concept; it is held in common.
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
- lazy caterer sequence - to three dimensions
related
- positive integer - in registry terms
- OEIS - as A000125
- combinatorial geometry - and recreational mathematics
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.cake-number
- Wikidata: Q4516348 (https://www.wikidata.org/wiki/Q4516348)
- OEIS: A000125 cake numbers
Direct properties not applicable Not applicable
- formula: (n^3 + 5n + 6) / 6 formula
- first terms: 1, 2, 4, 8, 15, 26, 42, 64, 93 sequence - OEIS A000125
- binomial form: C(n,3) + C(n,2) + C(n,1) + C(n,0) formula
Plane XCT: no invented physical properties.
Recognition optional Filled
- Maximum pieces from n planar cuts of a cube
- Sequence 1, 2, 4, 8, 15, 26, 42, 64
- The lazy caterer sequence is the two-dimensional case; powers of two count pieces only up to three cuts
- Not a visible object; a cube divided by planes.
Capabilities and actions required Filled
- explain the definition and formula
- relay the derivation
- describe puzzle uses
- distinguish related sequences
Hazards and failure modes required Filled
- Assuming pieces double with each cut
- Confusing with the lazy caterer sequence
- Requiring equal pieces, which the problem does not
Standards and interfaces required Filled
- No regulation; standard mathematical definitions
Context of use required Filled
- Not applicable; a mathematical concept.
- cake numbers for planes in three dimensions
- lazy caterer numbers for lines in two dimensions
- higher-dimensional generalisations by hyperplanes
- related problems such as cutting with restricted cuts
Sources Filled
- Wikidata item Q4516348: cake number - identity and sense of the item
- Wikipedia: Cake number - general description of the item
Open questions
- Should the lazy caterer sequence be a separate entry?
- How should OEIS and references be linked?
- How should higher-dimensional generalisations be recorded?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q4516348/spec.json