Ground Truth · a count pushed past the record

The Count That Ran Off the Page

Take a cube — n units on a side. Cut it, with perfectly straight slices, into smaller cubes with whole-number sides, filling it exactly. In how many different ways can it be done? A child can ask it. The answer is 1, 2, 10, 2098, 4 006 722 — and then it runs off the edge of what anyone had written down.

The two-dimensional version is old and famous — squaring the square. This is its cousin one dimension up. For a 1×1×1 cube there is exactly one way (leave it whole). For 2×2×2 there are two: the whole cube, or eight unit cubes. For 3×3×3 there are ten. You can see all ten yourself — below.

Instrument I — every way to cut the small cubes

cube side:

1 / 10

Every coloured block is one cube of the dissection; the whole thing fills the box with no gaps and no overlaps. Rotations and reflections count as different arrangements — the box sits fixed in space, so a tall cube in the front-left corner is a different dissection from the same cube in the back-right. (Count them by hand for n = 3 and you get ten. Good.)

Then it explodes

One more unit on a side and the count leaps in a way no amount of staring prepares you for. You cannot draw the n = 4 dissections one by one — there are 2098 of them. By n = 6 there are nearly three trillion. So don't take my word for the number: make your own browser enumerate them, right here.

Instrument II — count them yourself, live

press a button — the browser will enumerate every dissection

That last one is the point. Your own machine, running the same canonical enumeration the offline verifier runs, arrives at 2 954 374 781 704 — a number that, until it was computed for this page, appeared in no published record.

Off the edge of the record

The one place this sequence lives is a single line of commentary inside the Online Encyclopedia of Integer Sequences. Entry A228267 (R. J. Mathar and Rob Pratt, 2017) tabulates the counts for general a×b×c boxes, and notes, almost in passing: "The main diagonal T(n,n,n) is 1, 2, 10, 2098, 4006722, …." Five terms, then a dash. The diagonal is not a sequence of its own; nobody had recorded a sixth.

The diagonal D(n) — ways to cut an n×n×n cube into cubes

n	D(n)	× previous	scale (log)

The growth rate itself keeps accelerating: each term is a larger multiple of the one before. The sixth term computed here extends the record by one; the seventh, if it landed on the box this ran on, extends it by two and is a number with nineteen or twenty digits. Where exactly the count stops being reachable on ordinary hardware is stated honestly below — the wall is real, and named.

The check

Two independent methods agree. A memoized "skyline" count and a from-scratch enumerator that actually lays cubes into a 3-D grid give the identical count for every small box — the counting shortcut has to reproduce the brute force on its own.
The published record is reproduced term-for-term. The full A228267 triangle prefix (1,1,1,2,1,1,3,1,5,10,1,1,5,1,11,31,1,35,167,2098) and its anchors — T(2,2,r) is the Fibonacci numbers, T(3,3,r)=1,5,10,31,76,210, T(4,4,2)=35, T(4,4,3)=167 — all match.
The diagonal matches the five published terms, then continues: D(6)=2 954 374 781 704.
The gap is real. On 2026-07-08, both the diagonal 1,2,10,2098,4006722 and the value 2954374781704 returned no match from oeis.org — this term was recorded nowhere.
Everything here is recomputed live in your browser, and offline in research/cube-dissections/verify.mjs (18/18). The C++ solver that reaches the harder terms is committed beside it as cubes.cpp.