Stephen J. Pursey · Lakewood, Ohio

It takes eleven cubes to hide emptiness.

There are exactly 384 ways to do it. I counted them in 2026.

I hated math as a kid. Now I stay up late doing it for fun, answering questions nobody has answered and asking questions nobody has asked.

minimal cavity polycube · 11 cells · drag to rotate · A396101(11) = 384

11 OEIS entries touched 4 authored · 6 extended · 3 corrected

62 new terms computed 53 published · 9 in review

6 published errors corrected 5 live · 1 in review

6 original questions posed 4 answered · 2 open

Latest

authoredin review

A398967 · Gauss diagrams of meander curves, counted up to rotation

Companion to A399316 in which a diagram and its mirror image count separately: 1, 2, 6, 28, 166, 1141, 8645, 70351, 603253. Split out at an editor's request. Proposed Sep 13.

The full ledger

Briefly

Stephen J. Pursey, hobbyist mathematician, in the rose garden at the Biltmore Estate

Greetings from Lakewood, Ohio, on the shore of Lake Erie. By day I work as a supply-chain analyst. By night I do mathematics, and the questions I like best are the ones you can't look up anywhere, because nobody has ever answered them. So I answer them. Lately that means counting polycubes, especially the ones that enclose internal cavities (A357083, A355966, and A396101, that last one mine). When I'm not doing that, I play piano and run more home servers than any household needs.

All of this is a hobby. Amateur mathematics is a living tradition, and there's plenty of room in it for you.

The open questions

Try it yourself

Build your own eleven-cube

The amber cell in the middle wants out. Place cubes in the 3×3×3 frame until it can't reach the outside through any face, keeping everything you place connected as one piece. It takes at least 11 cubes. The theorem says you can't do better. The game is finding one at all.

0 cubes placed · 11 is the minimum

Solve it and a second prisoner arrives: a domino, ten doors, minimum fifteen. Its 4×3×3 arena is big enough, and I know that because I enumerated all 4,320 minimal domino builds and checked that every one fits inside it.

And the question that started everything: how many distinct minimal builds exist? (Fixed in space; rotations count separately.)