Don't step on the grass!

Geometry Level 3

In which figure is the black path the shortest ?

Difficulty: slightly spicy 🌶🌶🌶

This problem gets two chili peppers because there are some quick-and-clever solutions where the path lengths are carefully compared but never calculated, however most who guess at a glance guess wrong.

A B C Error: they are all the same length

1 solution

Marta Reece
May 10, 2017

Without calculations:

A: 9 quarter circles plus 6 radii

B: 4 quarter circles, but double sized, plus 2 doubles sized radii = 8 quarter circles original size plus 4 original radii

C: 2 quarter circles, quadruple sized, plus 2 quadruple sized radii = 8 quarter circles original size plus 8 original radii

Clearly B is the shortest. (8 < 9 and 4 < 6 < 8).

With calculations:

A: 9 quarter circles, radius $R=\frac{1}{8}$ , plus $6R$ is $(4.5\pi+6)R\approx2.517$

B: 4 quarter circles, radius $2R$ , plus $2\times2R$ is $(4\pi+4)R\approx2.071$

C: 2 quarter circles, radius $4R$ , plus $2\times4R$ is $(4\pi+8)R\approx2.571$

Don't step on the grass!

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