A square sheet of paper is folded in half along the vertical midline from left to right, and then folded in half again from top to bottom. A circular hole is punched through the center of the folded square, and a triangular notch is cut along the folded bottom-right corner. When the sheet is completely unfolded, how many circular holes and triangular cut-outs will appear on the entire sheet?

(A) 4 circular holes and 4 triangular cut-outs

(B) 4 circular holes and 1 diamond-shaped cut-out

(C) 1 circular hole and 4 diamond-shaped cut-outs

(D) 2 circular holes and 2 triangular cut-outs

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles Spatial aptitude paper folding tracking: - A square sheet folded twice produces 4 layers. - **Position of circular hole:** Punched through the interior (center) of the folded quadrant. Because it does not touch any folded crease edges, unfolding the 4 layers replicates the hole into exactly $1 \times 4 = 4$ separate circular holes symmetrically disposed in each quadrant. - **Position of corner notch:** Cut at the folded bottom-right corner, which corresponds to the central intersection of both main fold creases (the original center of the unfolded sheet). Cutting a triangle at this center vertex causes the 4 folded flaps to unfold into a single merged symmetrical diamond (rhombus) shaped cutout at the center of the sheet. ### 2. Analytical Formulation - Number of independent interior circles: 4 - Number of merged corner notches at the intersection of creases: $4 \times (1/4) = 1$ complete central diamond cutout. ### 3. Distractor Forensics - **Option A:** Fails to recognize that 4 corner triangular cuts around the common center vertex unite into a single continuous diamond hole. - **Option B:** Accurately identifies 4 individual circular holes and 1 central diamond cutout. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Cuts on fold creases merge across symmetric axes when unfolded; cuts away from creases duplicate identically. - **Standard Reference:** GATE General Aptitude Official Syllabus - Spatial Aptitude (Paper Folding and Cutting).

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