Which one of the following shapes can be used to tile (completely cover by
repeating) a flat plane, extending to infinity in all directions, without leaving any
empty spaces in between them? The copies of the shape used to tile are identical
and are not allowed to overlap.
Options:
(A) circle
(B) regular octagon
(C) regular pentagon
(D) rhombus
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.10)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *General Aptitude* | **Topic**: *Quantitative Aptitude* (`Numerical Computation & Estimation`)
- **Question Type**: **MCQ** | **Marks**: **2 Marks**
- **Quantitative / Analytical Framework**: Solved using principles of number theory, algebra, combinatorial counting, and spatial geometry.
**2. Step-by-Step Mathematical Derivation**:
- **Step 1 (Problem Translation)**: Extract all numeric constraints, boundary relations, and variable definitions from the given question.
- **Step 2 (Analytical Formulation)**: Apply the relevant algebraic identity, geometric theorem, or combinatorial relation.
- **Step 3 (Step-by-Step Execution)**: Solve the resulting system of equations to isolate the target variable, yielding **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `regular octagon`**.
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