Question

If you are asked to draw the following four figures with the following constraints: They must be drawn in a single stroke (without lifting the pen from paper), and each line is drawn only once. Which of the options can't be drawn? 

 

• A. A
• B. B
• C. C
• D. D

Hint:

For any "draw in one stroke" puzzle, immediately count the vertices with an odd number of lines connected to them. If that number is 0 or 2, it's possible. If it's any other number (like 4 in figures C and D), it's impossible. Be wary of questions in exams that may have incorrect keys.

Answer 1

The Correct Option is C, D

Step 1: Understanding the Concept: 
This problem is an application of graph theory, specifically related to Eulerian paths. A figure can be drawn in a single, continuous stroke without retracing lines if and only if its corresponding graph has either zero or exactly two vertices (nodes) of odd degree. The degree of a vertex is the number of lines meeting at that point. 

Step 2: Detailed Explanation: 
Let's analyze the degree of each vertex (represented by dots or line intersections) for all four figures. 
Figure A: There are 9 vertices.
- The central vertex has 4 lines meeting (degree 4).
- The 4 vertices on the diagonals have 4 lines meeting (degree 4).
- The 4 vertices on the outer square have 4 lines meeting (degree 4).
All 9 vertices have an even degree. A graph with zero odd-degree vertices has an Eulerian circuit and can be drawn in a single stroke. Therefore, Figure A is drawable.
Figure B: There are 9 vertices.
- The central vertex has 4 lines meeting (degree 4).
- The 4 vertices on the corners of the inner square have 4 lines meeting (degree 4).
- The 4 outer vertices where the loops meet have 4 lines meeting (degree 4). All 9 vertices have an even degree. Therefore, Figure B is also drawable.
Figure C: There are 8 vertices to consider.
- The 4 vertices at the corners of the square (the black dots) each have 3 lines meeting (2 from the square, 1 from the loop). Their degree is 3 (odd).
- The 4 points where the loops cross the square sides are vertices of degree 4. Since there are four vertices of odd degree, this figure cannot be drawn in a single stroke.
Figure D: There are 9 vertices to consider.
- The 4 vertices at the corners of the square (the black dots) each have 5 lines meeting (2 from the square, 1 from the diagonal, 2 from the loop passing through). Their degree is 5 (odd).
- The central vertex has 4 lines meeting (degree 4).
- The 4 vertices where the diagonals intersect the loops have degree 4. Since there are four vertices of odd degree, this figure cannot be drawn in a single stroke.
Step 3: Final Answer: 
The correct options are (C) and (D).