Two points, a straight riverbank, and one required stop. The optimal route appears when the problem stops looking like a detour.
The Stop at the River
Riddle statement
Two points, A and B, lie on the same side of a perfectly straight river.
You want to start at A, touch the river at some point to fill a bottle, and then reach B.
At which point on the river should you stop so that the total path is as short as possible?
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Solution
Answer: touch the river at the point where the line from A to the reflection of B across the river crosses the bank.
Reflect point B across the river, as if the river were a mirror. Call the reflected point B’.
Now draw a straight line from A to B’. That line crosses the river at one specific point.
That is the point where you should touch the river.
Why? Because any path from A to the river and then to B becomes, after reflecting B, a path from A to the river and then to B’. And the shortest path from A to B’ is a straight line.
So the optimal point is the one where the angle of approach to the river and the angle of departure toward B are equal, just as in a reflection.