GATE 2026 ME – Question 45
Consider the Euler variables of polyhedral objects namely, P1, P2, P3, and P4 as given in the table below.
| Polyhedral objects | Faces (F) | Edges (E) | Vertices (V) | Faces' inner loops (L) | Bodies (B) | Genus (G) |
|---|---|---|---|---|---|---|
| P1 | 6 | 12 | 8 | 0 | 1 | 0 |
| P2 | 5 | 8 | 5 | 0 | 1 | 0 |
| P3 | 5 | 12 | 8 | 0 | 1 | 0 |
| P4 | 10 | 24 | 16 | 0 | 1 | 0 |
Which one of the following options is NOT a topologically valid closed polyhedral object as per Euler's law?
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Correct answer: (A) P3
Explanation
For a closed polyhedron of genus 0 with one body and no inner loops, Euler's law says $V - E + F = 2$. P1 gives $8 - 12 + 6 = 2$, P2 gives $5 - 8 + 5 = 2$ and P4 gives $16 - 24 + 10 = 2$. P3 gives $8 - 12 + 5 = 1$, which is not 2, so P3 is not valid.