# Can this be drawn in one line without going over the line? Can this image be drawn in one line and without going over any lines?

• Has a correct answer been given? If so, please don't forget to $\color{green}{\checkmark \small\text{Accept}}$ it :) – Rubio Mar 8 at 21:17

No

Because

There should be at most $$2$$ intersections with odd number of passing lines. In the picture, there are $$8$$. More reference: here

The reason that it is impossible to draw the figure without picking up your pencil/redrawing over some lines is, as athin said, because there are more than two vertices with an odd number of "paths" leading to that vertex.

Here's why that is the case:

Imagine that a shape like the letter Y is in the image. The center of the Y is a vertex with three paths coming to it. In the presented problem, if the pencil passes through the point at the center of Y, it will cover exactly two paths. That leaves one path, still to be covered, that leads to nowhere. If all three paths are to be used, the center of the Y must be a starting or ending point in the solution. There is no other way to reach all three paths. The properties of Y discussed apply to any vertex with an odd number of paths leaving it.

Now picture an

X. Do you see how the point at the center of it has an even number (4) of paths that lead to it. This means that the pencil in the question can pass through X and leave 2 paths, allowing the pencil to pass through X yet again without stopping. This applies for all even numbers in the same way that Y's properties apply to all odd numbers - take an *, for example. See how it has an even number (6) of paths leading to it? The pencil can pass through once, leaving 4 paths, which we just showed can be passed through without stopping.

Bringing this all together,

Any vertex with an odd number of paths leading to it must be an ending or beginning point for the pencil. There can only be one beginning and one ending. Thus, as athin said, having more than two vertices with an odd number of paths results in a figure that is impossible to draw with one continuous stroke. Your figure has 8 vertices with an odd number of paths, and as I'm sure you are aware, 8 is more than 2.