Optimal Fence Riddle

A long time ago I found a website with a very interesting riddle. I haven’t been able to find the website again, but I thought I’d do my best to recreate the riddle from my memory and post it here.

You are a farmer that has a square field. Your field neighbors equal square-sized fields north, east, south and west:

Farmer Fence Riddle

Here is the problem. The neighboring fields have roaming cows that love to hang out with each other. The problem is that they keep crossing over your field in order to do so.

These cows, while social, will only visit a neighboring field if they can see those cows.

As an engineer, you decide to build a fence. So you build one like so:

Farm Riddle Fence Example

This X shaped fence prevents any of the cows from one neighboring to see any of the cows from another.

Here’s the question: Can you find a solution that uses less fence?

2 thoughts on “Optimal Fence Riddle”

  1. This particular problem is known as the Opaque Forest Problem on Wikipedia, and has been unsolved for almost 100 years.

    If you require the fences to connect, then there’s a known solution – the Steiner Tree – with length 1+sqrt(3) times the side of the square (compared to 2*sqrt(2) times the side of the square for the two diagonals).

    It’s known that the solution must have length at least 2, but the solution that’s generally accepted as probably being best has length sqrt(2) + sqrt(3/2) or ~2.639

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