I mean $ \sqrt{w^2+(21-x)^2}+\sqrt{(20-w)^2+z^2}+\sqrt{x^2+(20-y)^2}+\sqrt{y^2+(21-z)^2}.$

The command

`Minimize[Sqrt[x^2 + (20 - y)^2] + Sqrt[y^2 + (21 - z)^2] + Sqrt[z^2 + (20 - w)^2] + Sqrt[w^2 + (21 - x)^2], {x, y, z, w}] `

is running without any response on my comp for hours. The numerical optimizations

`NMinimize[ Sqrt[x^2 + (20 - y)^2] + Sqrt[y^2 + (21 - z)^2] + Sqrt[z^2 + (20 - w)^2] + Sqrt[w^2 + (21 - x)^2], {x, y, z, w}, Method -> "DifferentialEvolution"] `

`{58., {x -> 11.579, y -> 8.97237, z -> 11.579, w -> 8.97237}`

and the same with `Method->"RandomSearch"`

`{58., {x -> 10.5551, y -> 9.94753, z -> 10.5551, w -> 9.94753}}`

and the same with `Method->"NelderMead"`

`{58., {x -> 18.3218, y -> 2.55062, z -> 18.3218, w -> 2.55062}}`

suggest the optimal value under consideration is taken in many points.