Walking diagonally across a square lot saves the man about 30% of the distance compared to following the two edges. This classic geometry puzzle highlights the efficiency of the shortest path.

Quick Math Breakdown

Assume the square has side length xxx.

  • Edge path : Two sides, so 2x2x2x.
  • Diagonal path : By Pythagoras, x2≈1.414xx\sqrt{2}\approx 1.414xx2​≈1.414x.
  • Savings : 2x−x2=x(2−1.414)=0.586x2x-x\sqrt{2}=x(2-1.414)=0.586x2x−x2​=x(2−1.414)=0.586x.
  • Percent saved : 2−22×100%=(1−0.707)×100%≈29.3%\frac{2-\sqrt{2}}{2}\times 100%=(1-0.707)\times 100%\approx 29.3%22−2​​×100%=(1−0.707)×100%≈29.3%, or roughly 30%.

Why It Works

The diagonal is nature's shortcut—think hypotenuse in a right triangle formed by the sides. Real-world example: Crossing a 100m x 100m field diagonally (141m) beats 200m along edges, saving ~59m or 30%.

Variations in Puzzles

  • Some versions use rectangles (e.g., 3m x 4m), but this specifies a square.
  • Exact value: 100(1−22)%≈29.29%100(1-\frac{\sqrt{2}}{2})%\approx 29.29%100(1−22​​)%≈29.29%, always rounded to 30%.

TL;DR : Approximately 30% saved.

Information gathered from public forums or data available on the internet and portrayed here.