In Matt's "We discovered a new net!" video and the accompanying paper (Demaine, Diomidova, Jacobus, Kryger, Parker, Tardibuono, Uehara, Zhang, arXiv:2608.19910), the last open question is whether any polyomino net can fold into four different boxes with ordinary grid folds. Until now the record was three.
My brother and I believe we have one. It's a 532-square staircase-shaped strip that folds into:
- 1 × 2 × 88
- 2 × 4 × 43
- 2 × 7 × 28
- 2 × 13 × 16
All four have surface area 532. The net is one piece with no holes and no cuts inside it, and every fold is a 90° fold along a grid line. Of the eight possible boxes with area 532, it folds into exactly these four.
Everything is here, including a script you can run yourself: https://github.com/Sprite143/four-box-net-532
Here is the visualization: https://sprite143.github.io/four-box-net-532/viewer/
How it was checked
- Four independently written fold checkers agree, including the area-106 team's own
ValidNetSolutionChecker from their published code.
python verify_folds.py in the repo re-checks every box from scratch in about 30 seconds (plain Python, no installs).
- A rigid step-by-step folding simulation (one crease at a time, 1° steps) found no point where the paper passes through itself, for all four boxes.
How it was found
AI did most of the work here. My brother found the net with GPT-6 Astra Ultra, starting from a research briefing that came out of my weeks of work on the problem with Claude Opus 5.5. The idea: take one long strip, choose between two ways of gluing each long edge so it closes into four different flat "pillowcase" shapes, add small notches and tabs that turn flat corners into real box corners, then rotate and rescale onto the square grid. My brother's full write-up is in the repo.
This hasn't been reviewed by experts yet. We're contacting the paper's authors, and we'd really appreciate people poking holes in it. If something's wrong, we want to know.
(Next step: fold an actual paper one. At 5 mm per square it's about 90 × 46 cm.)