Here’s a quick challenge. Do you think you draw these two shapes without lifting your pen and without retracing any edge? Now's a good time to pause the video and try solving this.

If you tried the puzzle, you may have already figured out how to draw the first shape. If not, I’ll show you one possible solution shortly. The second shape, however, is a different story. I claim that it’s impossible to draw the second shape under the given rules. Let’s analyze both shapes in detail to see why.

For the first shape, I claimed that it can be drawn without lifting the pen and without retracing any edge. To prove this, let’s first label its vertices as A, B, C, D, and E. Now, follow this path: start at B → go to A → D → back to B → C → D → E → A → and finally end at C. This completes the drawing without breaking the rules, proving that the first shape is indeed drawable with given rules.

For the second shape, I claimed that it’s impossible to draw it without lifting the pen or retracing any edge. Before we begin the analysis, let’s label its vertices as A, B, C, and D.

To prove that it’s impossible to draw this shape under the given rules, we’ll examine the problem in two separate cases. In the first case, we’ll test whether it’s possible to start and end at the same vertex without lifting the pen or retracing any edge. In the second case, we’ll check if starting and ending at different vertices could work. These two cases together exhaust all possible ways of drawing the shape, so if both fail, it will confirm that the shape cannot be drawn within the given constraints.


Let’s look at case 1. Suppose we try to draw the shape starting and ending at vertex A. For example, we go from A to D, then to C, and back to A. In this path, each vertex we visit has an even degree — meaning it has an even number of connected edges. However, in the actual shape, every vertex has an odd degree. This means it’s impossible to draw the shape while starting and ending at the same vertex under the given rules.

For case 2, let’s try drawing the shape starting at vertex A and ending at vertex B. Suppose we go from A to D, then to C, and finally end at B. In this path, every vertex except the start and end points ends up with an even degree. This leads to a key observation: if a shape has more than two vertices of odd degree, it cannot be drawn under the given conditions. Looking back at the original shape, all four vertices have an odd number of edges. Therefore, no matter which vertices we choose to start and end at, the shape cannot be drawn without breaking the rules.

So, the analysis of two cases has proven that it is impossible to draw this shape without lifting the pen or retracing an edge.

Here are some additional details for those with a mathematical interest. In graph theory, the problem of drawing a shape without lifting the pen or retracing any edge is known as finding an Eulerian path, named after the mathematician Leonhard Euler, who established the conditions for its existence. These conditions are:
1) If all vertices in a connected graph have an even degree, the graph has an Eulerian path that starts and ends at the same vertex.
2) If exactly two vertices in a connected graph have an odd degree, the graph has an Eulerian path that starts at one of those odd-degree vertices and ends at the other.