soft-to-rigid-packing · cubes in cubes · n = 12

Twelve unit cubes fit in a cube of side 2.93151

Start with balls, squeeze them into a box, then harden them into cubes while the walls keep pushing. On 8 October 2026 this found a packing of 12 unit cubes 0.0012533 smaller than the July 2026 record. Everything needed to check it is below and in the repository.

Coordinates (JSON) Verify Paper PDF Repository

2.9315185094797side, s_full (Friedman display 2.93151+)

2.9327717687previous record, H. Lin, July 2026

−0.0012533improvement

1.0 × 10⁻⁶min gap between any two cubes and to every wall

7,700random starts run, squares and cubes

6published square records rediscovered from random starts

The two winners

n = 12 cubes, new record. Seed 52: balls compress, harden into cubes, settle; the last frame is the published claim file. Dashed: the previous record's box. Drag to rotate.

n = 11 squares, where it started. The first run to reach Trump's 1979 packing (side 3.877084, proved optimal in 2026) from random disks. Dashed: Trump's box.

Verify it

The claim is a list of twelve poses. Pose i is a centre c_i and a scalar-first unit quaternion q_i; cube i is the unit cube centred at c_i rotated by q_i. The container is [0, s]³ with s = 2.9315185094797.

Corners.

v_{i, g} = c_{i} + R (q_{i}) g, g \in {- \frac{1}{2}, \frac{1}{2}}^{3}

Inside the box. every coordinate of every corner satisfies

0 < v_{k} < s

. Minimum wall gap: 1.0000000000e-06.

No overlaps. Two convex boxes are disjoint iff some axis separates their projections. For cubes it suffices to test 15 axes: the 3 face normals of each cube and the 9 cross products of their edges. Pair gap = max over those axes of the gap between projection intervals. Minimum over all 66 pairs: 1.0000000006e-06.

Exactly. certify_exact.py converts every number in the file to a rational, uses the exactly orthogonal rotation

H (q) / {| q |}^{2}

, and exhibits a strictly separating plane for every pair with no rounding: CERTIFIED.

Run the same checks in this page:

Loads the JSON from this site and runs the 15-axis test on all 66 pairs.

Or from the repository:

git clone https://github.com/yoheinakajima/soft-to-rigid-packing.git && cd soft-to-rigid-packing
python3 verify.py claims/cubincub_n12/cubincub_n12.json        # standard library only, exits 1 on failure
python3 certify_exact.py claims/cubincub_n12/cubincub_n12.json # exact rational arithmetic
./scripts/reproduce_n12.sh                                      # rerun the search from seed 52

Same file format and checks as the current record file, Hyra-results/cubincub_n12.json, so the two can be compared line by line.

What we found

n = 12 cubes: a new record. 2 of 192 ball → cube starts land below the previous record, in two different arrangements (2.931515 and 2.931620 before adding clearance). The rigid-start control's best of 32 is 2.99257.
The method is reliable on squares. From random starts it reaches Trump (n = 11), Bidwell (17), Hämäläinen (18), Wainwright (19), Friedman (26) and Göbel (27) to 10⁻¹², and beats a rigid control run on the same seeds.
Starting soft is what matters. Rigid cubes or squares started from the same positions jam. Starting from balls, the cubes are already densely packed before orientation matters.
Where it fails. No square record was improved. Squares n = 28 and 29 stay 0.02 and 0.005 above the record; cubes n = 10 and 11 settle in known but worse packings. An octagon-shaped intermediate, meant to give rotation a signal earlier, did not help.

How we got here

7 Oct · squares, n = 11
Can blobs that harden find the best packing?
Yes: 6 of 160 starts reach Trump's packing; 0 of 120 rigid starts do.
7 Oct · squares, n = 2–30
Same settings, every n
Tilted records found up to n = 11, none beyond. Rigid starts win when a grid has empty cells; blobs win when it is full.
7 Oct · squares, n = 17 and 19
Five setups on the same seeds
Blobs reach Bidwell's n = 17 packing 10 times in 960 starts; rigid and octagon starts never do.
7 Oct · exact tightening
From penalty slack to exact contact
A constrained solver (corners in the box, a separating plane per pair) snaps near-misses onto the records to 10⁻¹².
8 Oct · squares, n = 26–29
More hardening time
At n = 26 the hit rate climbs from 2.5% to 10.4% as steps grow; n = 28 and 29 stay out of reach.
8 Oct · cubes, n = 9–12
Same method, one dimension up
Friedman's n = 9 packing is found 16 times in 64. At n = 12, seed 52 lands below the July 2026 record.
8 Oct · claim
Certified and published
Clearance of 10⁻⁶ added, exact certificate passed, file in the record's format. Submitted to Erich Friedman's catalogue: pending.

Every experiment, including dead ends, is in EXPERIMENT_LOG.md.

Paper

Twelve Unit Cubes in a Cube of Side 2.9315, Found by Hardening Balls into Cubes. Read it online, with the interactive 3D figure, or as a PDF. Both are rebuilt from paper/main.tex on every commit, so these links always point to the latest draft.

built from commit efe7a92 · 2026-10-08 21:39 UTC

The two winners

Verify it

What we found

How we got here

Can blobs that harden find the best packing?

Same settings, every n

Five setups on the same seeds

From penalty slack to exact contact

More hardening time

Same method, one dimension up

Certified and published

Paper