Erik Demaine and Martin Demaine's Puzzles:

Cube Folding Puzzles

What polyominoes fold into a cube? This seemingly simple question is still an unsolved mathematical problem. These puzzles illustrate some of the cases we (only recently) understand.

These cube folding puzzles are based on the following sources:

  1. Martin Gardner, “Paper cutting”, in New Mathematical Diversions, Revised, Mathematical Association of America, 1995.
  2. Jill Bigley Dunham and Gwyneth R. Whieldon, “Enumeration of solutions to Gardner's paper cutting and folding problem”, in The Mathematics of Various Entertaining Subjects, volume 2, 2018, pages 108–124.
  3. Oswin Aichholzer, Michael Biro, Erik D. Demaine, Martin L. Demaine, David Eppstein, Sándor P. Fekete, Adam Hesterberg, Irina Kostitsyna, and Christiane Schmidt, “Folding polyominoes into (poly)cubes”, International Journal of Computational Geometry and Applications, 28(3):197–226, 2018.
  4. Oswin Aichholzer, Hugo A. Akitaya, Kenneth C. Cheung, Erik D. Demaine, Martin L. Demaine, Sándor Fekete, Linda Kleist, Irina Kostitsyna, Maarten Löffler, Zuzana Masárová, Klara Mundilova, and Christiane Schmidt, “Folding Polyominoes with Holes into a Cube”, in Proceedings of the 31st Canadian Conference in Computational Geometry, August 2019.

PDFs to print

To try these puzzles, print out one of the 11-page PDFs, cut out each puzzle you'd like to try, cut along the indicated slits (doubled bold lines), and cut out any gray squares with a scissors icon. You can choose from two editions of the puzzle:

MoMath edition, from a Cube Conundrums workshop in The Six Series MathFest 2019 edition
OrigaMIT 2019 edition PCOC 2021 edition

Last updated June 11, 2021 by Erik Demaine.Accessibility