What is the smallest polycube whose cavity has two cells, and how many minimal examples are there?
15 cells. Exactly 4320 fixed examples; 180 up to rotation and reflection.
Minimality and census both proven in the campaign-one paper.
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Questions nobody had asked
A question can be original the same way a theorem can. These are mine — with their status, and their answers as they land.
These are free. Steal them. Cite generously.
What is the smallest polycube containing two distinct cavities?
17 cells — and there are exactly 369 minimal examples up to rotation and reflection (718 one-sided, 16,968 fixed).
Census of all 206,155,755 size-16 cavity polycubes: none has two cavities. A targeted search over anchored cavity pairs then found the 17-cell examples — the two sealed cells either diagonally adjacent or collinear at distance two, sharing shell walls — and ruled out every other geometry. The full census at 17 followed: every example verified by flood-fill, and the fixed total confirmed by two independent derivations.
For a polycube shape R, let f(R) be the minimum number of cells enclosing a cavity of shape R. Is f monotone under sub-shapes? What is its growth rate? Which shapes are realizable as cavities at all?
Not monotone: shrinking the cavity you want can strictly raise the price of building it. Realizable exactly when the shape does not enclose a cavity of its own. Growth ranges from Θ(n^(2/3)) for compact shapes to Θ(n) for rods.
The counterexample is exact: remove one face-center cell from a 3×3×3 cavity and the minimum enclosure rises by precisely one. Underneath, f(R) is a Steiner connection problem on the lattice outside R. Known values, each a small theorem: f(cell) = 11, f(domino) = 15, f(2×2 square) = 21, f(tripod) = 22, f(plus-pentomino) = 27. Full writeup is folding into paper two, together with a new sequence: the smallest polycube admitting an n-cell cavity, the 3D analogue of A283056.
In d dimensions, what is the minimum hypercube count enclosing a void, and how many minimal enclosures exist?
Pattern so far: d = 2 gives 7 (classical); d = 3 gives 11, with exactly 384 minimal enclosures — the spanning trees of the octahedron. Conjecture: the minimum is 4d − 1 in every dimension, with the count given by spanning trees of the d-dimensional cross-polytope graph. If true, the cost of hiding emptiness grows linearly in dimension, and the ways to do it minimally are counted by Kirchhoff. Novelty check against the folklore literature in progress.
answered this? send it in →How many n-cell polycubes have exactly k cavities? How many by cavity volume? By cavity shape class?
First terms computable from enumeration dumps already generated; candidate new OEIS sequences.
answered this? send it in →Is the oriented-CI classification for cyclic groups exactly the k, 2k, 4k list with no exceptions?
Our data: Z8, Z9, Z18 — precisely the exceptional orders of the undirected classification — all fail CI for oriented graphs. If the oriented case differs from the digraph case anywhere, that gap is publishable. Literature check in progress.
answered this? send it in →The adopted column
Questions others asked
Standing questions from the literature — flagged in OEIS entries, left open in comments, waiting sometimes for years. I collect them here and chip away. The answered ones keep their receipts.
Two published counts of oriented circulant graphs disagree. Can anyone reproduce the older table's value a(8) = 7?
No — because it is wrong. The true value is 9, and the published table is also wrong at n = 12 and n = 15 (true values 70 and 290). Howroyd's own computation was correct everywhere.
Exhaustive isomorphism classification, verified three independent ways, plus five new terms a(16)–a(20). The wrong values trace to a table in the 2001 source paper. Bonus finding: the cyclic groups of order 8, 9, and 18 — a known exceptional family — fail the Cayley-isomorphism property for oriented graphs.
What are the next terms of A357083 — free polycubes of size n whose complement is not connected?
a(15) = 422,277; a(16) = 4,310,738; a(17) = 41,982,903; a(18) = 395,335,115.
Exhaustive enumeration with a proven pruning threshold, anchored against the known totals of all fixed polycubes at every size.
What are the next terms of A355966? And were the published values even right?
They were not: a(14) = 76,017 (published 75,917) and a(15) = 838,575 (published 835,491). Plus three new terms through n = 18.
Two independent methods agreed with each other and disagreed with the published values; a full autopsy of the original program then accounted for both deficits to the exact object. The corrections are live and the faulty program was removed.
The entry A079815 contains a term whose origin its own contributor could not identify: "I don't know where the term 71 comes from."
Recomputable with standard graph-generation tooling. Queued.
answered this? send it in →A239231's terms (maximum values in Heyawake puzzles) have been corrected twice already. Are the current ones right?
A sequence that has been wrong twice deserves a third, independent look. Exact-solver verification plan drafted.
answered this? send it in →A170876, a three-dimensional toothpick sequence, carries its contributor's own warning — and disagrees with a companion entry.
A live contradiction between two published entries; at least one is wrong. Queued.
answered this? send it in →A288760's latest term is flagged by its own author as needing independent verification — and its predecessor was already retracted once.
Needs serious memory to verify; sized for the next hardware step.
answered this? send it in →The entry for snake polyominoes (A357234) carries a bare in-entry conjecture — a proposed closed form, posted as a question, awaiting proof or a counterexample.
Testable by exhaustive search at the next several sizes; an elementary proof looks plausible.
answered this? send it in →How many knights does it take to cover an n×n chessboard? For n = 22 through 27, the published values have been upper bounds — never proven optimal — since 2002.
Closing a published bound is the strongest kind of correction. Exact optimization does it.
answered this? send it in →How many ways can you fold an n×n map? Every published value descends from one program lineage — a translation of Lunnon's 1971 pseudo-code. A bug in the ancestor would live invisibly in every descendant.
Needs a genuinely independent algorithm, not another translation. Gardner-adjacent pedigree.
answered this? send it in →Counts of graphs whose symmetry group has exactly n elements (A080803) were sourced from a USENET thread and carry the entry's own warning that they still need verification.
Standard graph-generation tooling settles it tier by tier.
answered this? send it in →Queens domination counts (A075458): the hardest published value rests on a single PhD thesis's non-existence proof that no one has independently reproduced.
A machine-checkable certificate of the non-existence half would settle it permanently.
answered this? send it in →Several puzzle sequences were counted by hand and say so — 'should be verified' — in their own entries (A336660, A256641 among them).
Afternoon-scale each. Hand counts have a base rate, and it is not zero.
answered this? send it in →On Erdős problem #273, a forum comment explicitly asks for independent checking of a claimed nonexistence result.
A direct request, publicly posted, unanswered. Our favorite kind.
answered this? send it in →Kéri's covering-code tables — the reference records an entire area cites — have vanished from the web. Every link is a 404. Who keeps the records now?
Rebuilding and maintaining the tables makes the maintainer the area's record-keeper.
answered this? send it in →