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.
Original questions
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. There are exactly 369 minimal examples up to rotation and reflection (718 one-sided, 16,968 fixed).
A census of all 206,155,755 size-16 cavity polycubes turned up none with two cavities. A targeted search over anchored cavity pairs then found the 17-cell examples (the two sealed cells are 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?
Minimum 4d − 1 in every dimension, and the minimal enclosures are exactly the spanning trees of the cross-polytope graph, so there are (2d)^(d−2)·(2d−2)^d of them: 4, 384, 82944, 32768000, … (A193130). Up to symmetry the count is the number of nets of the d-dimensional hypercube, 1, 11, 261, 9694, … (A091159): the essentially different cheapest cages around a cell are the unfoldings of the d-cube.
Proof in the note linked below. The one-cell cavity forces its 2d face-neighbours, which are pairwise non-adjacent; sorting the remaining cells by distance from the cavity shows that at least 2d − 1 more are needed, with equality exactly when the extra cells are corner cells forming a spanning tree of the cross-polytope graph. A cavity of two or more cells always costs at least 4d cells. The count is the complete-multipartite spanning-tree formula. Checked by exhaustive search for d = 2, 3, 4 (7, 11, 15 cells; 4, 384, 82944 enclosures) and the count 32768000 for d = 5; the free counts 1, 11, 261 were checked by direct orbit counting. Both sequences were already in OEIS, neither with this interpretation.
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?
In my data, Z8, Z9, and 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 workbench
The hunt
These are the ones I'm actively hunting. No deadlines. Some of them may take years, and that's fine; it's a hobby.
The strong-coupling series for SU(2) lattice gauge theory stops where Münster left it in 1981, forty-five years ago. I want the next coefficient. The bottleneck is an enumeration engine, and enumeration is my home turf. This is the long one. It's one brick in a very large wall, but nobody has laid a brick there since before I was born.
The cavity-polycube counts end at n = 18 because that's where I stopped; the mathematics keeps going. A new algorithm, roughly twenty times faster, is built and proven correct. The run itself is waiting on a bigger machine. Then n = 19.
Cavity polycubes multiply roughly ninefold with each added cube at the sizes my table reaches, and that ratio is still descending toward a limiting constant nobody knows exactly. Rigorous bounds on growth rates of this kind are famously loose. Tightening one is a real theorem, and it would govern every sequence I've published.
About all of this: how a supply-chain analyst from Lakewood ended up correcting the mathematical record. It gets written after a few more campaigns, once there's an ending worth writing.
From the literature
Questions others asked
Standing questions from the literature: flagged in OEIS entries, left open in comments, some of them waiting for years. I collect them here and chip away at them. The answered ones keep their receipts.
Take a polyhedron in which exactly three faces meet at every corner, and mark each edge convex or concave. Which cubic graphs with such markings can occur? Knuth built a catalog to eight vertices, asked for a proof that K3,3 and one marking of the triangular prism are impossible, and asked whether the catalog is complete.
K3,3 and the questioned prism marking are impossible, the catalog is one short (the lopped prism has a fifteenth marking, which I found and verified in exact arithmetic), and the smallest marking that cannot be drawn flat has exactly fourteen corners, the number he conjectured.
Ten short lemmas about the shape at a corner, the convex hull, the face planes, and the line where two faces meet cut the possibilities down to a finite list. The one that settles the ten and twelve corner cases says the corners on such a line pair up into edges, so there is an even number of them. Two of his five questions were already answered by Peter Weigel in 2023; of the three still open, this closes one.
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, the 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."
Checked Sep 13, 2026: under the stated definition a(8) = 51, the same as A000512. McKay's suggested explanation, matrices with sorted rows and columns, gives 3 or 5 at n = 5 where the entry has 2, so 71 fits neither reading. The correction is in editorial review.
answered this? send it in →A239231's terms (maximum values in Heyawake puzzles) have been corrected twice already. Are the current ones right?
No. The entry gives a(13) = 56, and the true value is 57: a 57-cell solution exists and has been checked independently, and a SAT solver proves 58 impossible.
Every other term through n = 15 is confirmed exact by the same solver. n = 16 to 20 are being checked so the correction can go in as one edit.
A170876, a three-dimensional toothpick sequence, carries its contributor's own warning, and it 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 conjecture: a proposed closed form, posted as a question, still waiting for a 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 are upper bounds, and nobody has proven them optimal since 2002.
Exact optimization could close them for good.
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 sit invisibly in every descendant.
Needs a genuinely independent algorithm rather than another translation. Martin Gardner wrote about this family of problems.
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 in their own entries ('should be verified'; A336660 and A256641 among them).
Each is an afternoon of work, and hand counts do get things wrong.
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, and still unanswered.
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?
Whoever rebuilds and hosts the tables becomes the area's record-keeper.
answered this? send it in →