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.

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.

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 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 extends it by two and is a number with twenty-one digits, 159 536 323 338 948 052 780, computed by cubes.cpp and recorded in research/cube-dissections/data.json. Corrected 2026-08-02: this sentence still said "nineteen or twenty digits", written before D(7) had actually been computed and left stale once it had. Where exactly the count stops being reachable on ordinary hardware is stated honestly below — the wall is real, and named.