ExampleConstruction: AI-generatedVerification: Not suppliedSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Colourings of the faces of a cube up to rotation

Example

The $24$ rotational symmetries of the cube act on its $6$ faces with cycle types:

$1$ identity of type $1^{6}$ ;
$6$ quarter-turns of type $1^{2} 4$ ;
$3$ half-turns through opposite faces of type $1^{2} 2^{2}$ ;
$8$ third-turns through opposite vertices of type $3^{2}$ ;
$6$ half-turns through opposite edges of type $2^{3}$ .

Therefore the number of face-colourings up to rotation with $m$ colours is

$\frac{m^{6} + 3 m^{4} + 12 m^{3} + 8 m^{2}}{24} .$

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.