Question

Which of the given figures can be drawn without lifting the pen? The lines can cross each other but cannot overlap on top of one another.

• A.

  

• B.

  

• C.

  

• D.

  

Answer 1

The Correct Option is A, B, D

To solve the problem of determining which figures can be drawn without lifting the pen, we need to apply Euler's theorem on traversable networks. For a figure (graph) to be traversed without lifting the pen and without overlapping a line, it must satisfy the condition of having zero or two vertices of odd degree. Below is an analysis based on the provided figures:

• Fig 1: To check if it can be drawn without lifting the pen, count the number of edges connected to each vertex. Every vertex must have an even degree or exactly two can have an odd degree. Since Fig 1 fits this condition, it can be drawn as required.
• Fig 2: Similar analysis shows that this figure also satisfies the condition for being drawn without lifting the pen, as it has zero or an even number of odd-degree vertices.
• Fig 3: This figure does not satisfy the necessary conditions as it has more than two odd-degree vertices, which means it requires lifting the pen to draw without returning to the starting point.
• Fig 4: This figure can be analyzed similarly to determine that it meets the required conditions for traversal (either having no vertices of odd degree or exactly two).

Based on this analysis, the figures that can be drawn without lifting the pen are:

,

,