Question

Which of the given figures can be drawn in one stroke without lifting the pen or retracing any line? 

• A. None of the figures
• B. Figures I, II, and III
• C. Figure IV
• D. All of the figures

Hint:

Count odd-degree vertices: \(0⇒\) closed tour, \(2⇒\) open tour, \(>2⇒\) impossible.

Answer 1

The Correct Option is B, C, D

Model each drawing as a graph (intersections = vertices, segments = edges). A connected graph has a one-stroke drawing (Euler trail) iff it has \(0\) or \(2\) vertices of odd degree.
Figure I: Has \(>2\) odd-degree vertices \(⇒\) not possible.
Figure II: Exactly \(2\) odd-degree vertices \(⇒\) possible (open trail).
Figure III: Exactly \(2\) odd-degree vertices \(⇒\) possible (open trail).
Figure IV: All vertices even \(⇒\) Euler circuit exists (closed trail).
Therefore, the drawable figures are \(\boxed{B, C, D}\).