A 4D search fit 26 to 30 unit hypercubes in smaller boxes than any construction we could find. You can rerun it right here.
Interactive
Drag to turn it in 4D. Press Run search to watch a packing squeeze itself together.
Small squares: the packing's shadows on the six coordinate planes, box drawn as a square.
Loading the found packing.
Run builds a fresh packing from the seed; Basin hop tries to shrink whatever is on screen.
The 25-cube construction is stretched to side s₀ and n − 25 cubes are dropped in at random; then the box shrinks under no-overlap constraints. The same seed and settings give the same run, though browsers can round differently late in a run. With the defaults, seed 23 settles near 2.9435 for n = 26.
The short version
How small can a 4-cube be if it must hold n unit 4-cubes, turned any way you like? Side 2 holds 16 and side 3 holds 81; in between, the cubes have to tilt.
Turn the middle square 45° and five fit in side 2 + 1/√2. Do it in (x₁, x₂) and (x₃, x₄) at once: 5 × 5 = 25 tesseracts.
Erich Friedman catalogues this puzzle in 2D and 3D, and records still fall: H. Lin's 3D entries for 11 and 12 cubes date from July 2026. The 5 × 5 product gives 25 tesseracts at side 2.7071. Stacking a 3D packing in two layers along the fourth axis gives more: Friedman's 13-cube record (2.956) makes 26 cubes and his 14-cube record (2 + 7√2/10 ≈ 2.98995) makes 28. So after 25 the side jumped to 2.956, then 2.98995, then 3. The new packings land in that gap.
Sixteen cubes sit nearly axis-aligned in the corners. The rest fill the channels between them, each turned 25°–45° within one coordinate plane; one is turned in two planes at once, the 4D cousin of the 45° centre square. The corner frame opens up by about 0.24 compared with the 25-cube construction. Cubes beyond 25 sit in channels that mix one middle coordinate from (x₁,x₂) with one from (x₃,x₄), turned 45°. Four of them fit at about 2.9428 (n = 29); a fifth pushes the side to 2.9896 (n = 30).
Each pair of cubes carries its own separating direction as a free variable, so a positive gap along it proves the pair does not overlap. The C++ polish has hand-written gradients and reaches 2 + 1/√2 on the 5-square test to 10⁻¹⁰ in a third of a second; this page runs a line-for-line JavaScript port. Random starts never beat the constructions in 4D. Seeded starts gave 2.943189 (n = 26) and 2.951791 (n = 27) in the first batch of 20 seeds, and basin hopping brought both to 2.942809. For n = 28 to 30, seeds that put the extra cubes straight into those channels, one per arrangement up to the frame's 128 symmetries, did far better than random ones.
An independent NumPy checker tests every pair of cubes against all 56 candidate separating directions of the 4D separating-axis theorem, then the walls and the orthogonality of each rotation. Smallest gap: 4.5 × 10⁻¹¹ (n = 26), 1.05 × 10⁻¹⁰ (n = 27), 4.2 × 10⁻¹¹ (n = 28), 3.6 × 10⁻¹⁰ (n = 29) and 2.8 × 10⁻¹¹ (n = 30). Walls are exact to rounding; rotations are orthogonal to 4 × 10⁻¹⁶. The sides match 2 + 2√2/3 to 4 × 10⁻¹⁰ (n = 26) and 2 × 10⁻⁷ (n = 27). Still missing: an exact certificate with a safety margin, like the one Nakajima produced for his 3D claim.
Leaderboard
Smallest known side s(n) for n unit 4-cubes. Rows 26 to 30 are our candidates; the rest are built from lower-dimensional results. Our searches for n = 31 and 32 have not gone below 3.
| n | side s(n) | how | source | status |
|---|---|---|---|---|
| 1 | 1 | one cube | trivial | optimal |
| 2–16 | 2 | 2×2×2×2 grid | trivial; no 4D optimality proof found (2D: s(2) = s(3) = 2 in Friedman's survey) | believed optimal |
| 17–25 | 2.7071072 + 1/√2 | 5-square packing in (x₁,x₂) × the same in (x₃,x₄) | 2D value: Friedman, Squares in Squares; our n = 17, 18 searches reach it, nothing lower | construction |
| 26 | 2.942809≈ 2 + 2√2/3 | 16 corner cubes + 10 tilted | this page; was 2.956, the 3D 13-cube record (Friedman 1998) in 2 layers | new candidate |
| 27 | 2.942809≈ 2 + 2√2/3 | 16 corner cubes + 11 tilted | this page; was 2.98995, the 3D 14-cube record (Friedman 1998) in 2 layers | new candidate |
| 28 | 2.942826≈ 2 + 2√2/3? | 16 corner cubes + 12 tilted | this page; was 2.98995, the 3D 14-cube record (Friedman 1998) in 2 layers | new candidate |
| 29 | 2.942834≈ 2 + 2√2/3? | 25-cube frame + 4 cross-channel cubes | this page; was 3, the 3×3×3×3 grid | new candidate |
| 30 | 2.989568 | 25-cube frame + 5 cross-channel cubes | this page; was 3, the 3×3×3×3 grid | new candidate |
| 31–81 | 3 | 3×3×3×3 grid | trivial; our n = 31, 32 searches found nothing below 3 | construction |
Best known sides for the lower-dimensional packings used above.
| dim | n | side | by | source |
|---|---|---|---|---|
| 2D | 5 | 2.7071072 + 1/√2 | classical, proved optimal | Friedman, Packing Unit Squares in Squares (EJC survey DS7) |
| 3D | 9–10 | 2.707107 | E. Friedman, 1998 | Friedman, Cubes in Cubes |
| 3D | 11 | 2.88295 | H. Lin (Tencent Hyra), July 2026 | Hyra results · announcement on X |
| 3D | 11 | 2.89444earlier | Berthold, Kamp, Mexi, Pokutta, Pólik, 2026 | Out-of-the-Box Global Optimization for Packing Problems (arXiv) |
| 3D | 12 | 2.93277 | H. Lin (Tencent Hyra), July 2026 | Friedman, Cubes in Cubes |
| 3D | 12 | 2.93152claimed | Y. Nakajima, 2026, self-reported | soft-to-rigid-packing repository |
| 3D | 13 | 2.956 | E. Friedman, 1998 | Friedman, Cubes in Cubes |
| 3D | 14 | 2.9899492 + 7√2/10 | E. Friedman, 1998 | Friedman, Cubes in Cubes |
4D jamming of spheres is a different problem with no cube values: Skoge, Donev, Stillinger and Torquato (2006).
Data
Each cube is a centre c ∈ ℝ⁴ and a 4×4 orthogonal R whose columns are its edges: vertices c + R·(±½, ±½, ±½, ±½), box [0, s]⁴, R row-major.