The milkmaid's shortest path from stool to stream to barn
A milkmaid starts at a stool, must walk to a straight stream to fill her pail, then carry it to a barn door. The stream doesn't lie directly between them — it's off to one side. She wants the shortest possible route: stool to some point on the bank, then bank to barn. Walking straight at the stream, then straight at the barn, is not the shortest way. Where should she touch the water?
Reveal the answer
Reflect the barn's position across the line of the stream to get a mirror-image point. The shortest route is the straight line from the stool to that mirror point — where this line crosses the stream is exactly where she should fill her pail. This 'reflection trick' (the same principle behind billiard bank shots and light bouncing off a mirror) turns a bent-path minimization problem into simple straight-line geometry. It's puzzle master Henry Dudeney's 'Milkmaid Puzzle,' from his 1917 collection Amusements in Mathematics.
— Henry Dudeney, Amusements in Mathematics — 1917, 'The Milkmaid Puzzle'
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