A labourer digs until his head is twice as far below ground as it now stands above
Professor Rackbrane meets a labourer, 5 feet 10 inches tall, standing in a hole he's digging. The labourer says he plans to keep digging until his head ends up exactly twice as far below ground level as it now stands above it. How deep will the finished hole be?
Reveal the answer
105 inches, or 8 feet 9 inches. If he's dug down d inches so far, his head stands (70 - d) inches above ground now. He'll stop once his final depth, 2d, leaves his head (2d - 70) inches below ground, and the puzzle sets that equal to twice his current height above ground: 2d - 70 = 2(70 - d), which gives d = 52.5 and a final depth of 105 inches. This is Puzzle No. 100 from Henry Dudeney's 1917 Amusements in Mathematics.
— Henry Ernest Dudeney, Amusements in Mathematics — Puzzle No. 100, 1917