An infinite walk in $\mathbb{N}^{16}$, using only unit steps, with no three collinear points

2026-09-05

Abstract

Is it possible to walk to infinity, avoiding three collinear points, using as steps only the standard unit basis vectors $(0,0,\ldots, 0, 1, 0, \ldots, 0)$? We prove this is possible in $16$ dimensions.

Problem

• infinite walk
• collinear points
• unit steps

Innovation

• infinite walk
• collinear points
• unit steps
• 16 dimensions