But one of his students soon proved him wrong: Robert Berger found an aperiodic set of 20,426 tiles, and then a set of just 104 shapes. The game was on: How few shapes would suffice? Researchers ...

One set of numbers ... was answered only after the invention of algebra and a deeper understanding of π — the ratio of the circumference of any circle to its diameter. What does it mean to discover a ...

For centuries, one of algebra’s oldest puzzles has remained ... The trick lies in their use of combinatorics—the study of how elements in a set can be arranged or grouped.

The other proof relied on the team's discovery that there wasn't just one of these tiles, but an infinite set of related shapes that could all do the trick. The team's paper is available on the ...