Question

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?

• A. P3
• B. P1
• C. P4
• D. P2

Hint:

Check each object against the Euler-Poincare formula \(F-E+V=2(B-G)+L\) before picking the odd one out.

Answer 1

The Correct Option is A

Step 1: Rearrange Euler's formula to predict the edge count.
With one body, no holes, and no inner loops on any face, Euler's relation $F - E + V = 2$ can be rewritten as $E = F + V - 2$. Instead of computing $F-E+V$ directly, this step checks whether the reported edge count $E$ matches what $F$ and $V$ demand.

Step 2: Test P1.
Predicted $E = 6 + 8 - 2 = 12$. The table lists $E = 12$, so P1 is consistent.

Step 3: Test P2.
Predicted $E = 5 + 5 - 2 = 8$. The table lists $E = 8$, so P2 is fine too.

Step 4: Test P3.
Predicted $E = 5 + 8 - 2 = 11$. The table lists $E = 12$ instead, one edge more than a valid solid with those face and vertex counts could have.

Step 5: Test P4.
Predicted $E = 10 + 16 - 2 = 24$. The table lists $E = 24$, matching exactly.

Final Answer:
Only P3 has an edge count that disagrees with what Euler's formula demands, so it cannot represent a genuine closed polyhedron. \[ \boxed{P3} \]