Euler Trail

Problem:

Can you draw a continuous curve that crosses every line only once?



To make the word "line" clear i have marked below the lines that can be crossed. There are 16 of them.

So, can you do it?

One attempt can look something like the following:

In the above attempt although it crosses off most of the lines, it misses 1 of them. Can you cross every lines?



Give it a try!!




SPOILER ALERT!!!

Solution:

This problem can be solved in different ways. Your approach can differ from mine.

Lets look at these three structures which are parts of the original figure.

I have picked them particularly because they are unique. They have odd number of lines. i.e 5.


If i start drawing the curve from outside one of these structures then in order to finish crossing all the lines, ill end up inside the structure when i finish marking all the lines or if i start drawing the curve from inside then the final endpoint after crossing all the lines will fall outside the figure. In other words endpoint and startpoint cannot both fall in the same side.

The reason is that we have odd number of lines.



So, knowing this property, now we are ready to tackle the problem.

You will be entering two of these structures from outside. I hope you can see why. There are three of them and even if i start the curve from inside one of these, i will still be entering the rest two from outside. Now, we already saw above if we start from outside then when i finish crossing all the lines, ill be inside one of these from where i cannot move outside.  This is a problem. This also means we finish marking all the lines of one of these structures by entering inside the structure. But, wait!!! there are at least two of these. How can i be inside two of these at the same time if i want to finish marking all of their lines?

That's why its impossible to solve this puzzle.


Stay tuned for more fun puzzles!