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Jakob Steiner: 2 puzzles worth knowing

2 puzzles from Jakob Steiner, each explained in a minute and credited to the original work. Free on Savvy.

• Ring a gap between two circles with smaller circles — does the chain always close?

  Draw one large circle and a smaller circle fully inside it, off-centre. Now try fitting a chain of even smaller circles into the ring-shaped gap between them, each one touching its…

  — Jakob Steiner, Steiner Chain

• Connect three towns with the shortest road — the obvious answer isn't optimal

  You need to link three towns with roads so all three are connected, using the least total length of road. Simply drawing a road between every pair works, but it isn't optimal. A si…

  — Jakob Steiner, Steiner tree problem

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