Ideas for ‘The Shortest Road Track’ Problem

See the Problem here.

Here are some of the ideas received for the problem.

Discussion

Let us assume that the square has a side of 1 unit length. After some thinking, it becomes quite visible that curved lines would be of no use in this problem. This is because any 2 points joined by a curved line would be better off with a straight line.

As we start thinking about the problem, we see that the 4 corners can be connected to each other using three lines as shown below. In this case, the total length will be equal to 3 units.

The same length can also be achieved in a different way as shown below. Here, we have added 2 new junctions, where 2 points each meet.

Another way could be to join the diagonals. A diagonal is roughly 1.41 times the side of a square. So joining the 2 diagonals making the total length equal to 2.82 units. Here we have added another 1 junction where 4 points meet.

Now the question that arises is, what happens if we try to find a middle way among the above 2 images. Look at the series of images below and try to see what’s happening. How is the total length being affected?

As we split the middle point into 2 points and move them apart, what is happening?

• Lengths of the 4 end points joining the new junctions are decreasing
• There’s an additional line which gets added in the middle

We know that total length at the exact middle is 2.82 units and the extremes is 3 units. The following questions occur now.

Is the total length always increasing?
If not, Where is the optimal point where the length is shortest?

Further discussion to follow. Thanks to all those who have contributed their ideas. Go to the problem page, and submit your amazing ideas, which can take this discussion forward.

Or look at the Problems Page for more problems to explore.