From a square metal plate of side 2d, circular plates of diameter d are cut without overlap. What is the maximum number of such circles that can be cut with minimum material loss?

Introduction / Context:
To minimize material loss when cutting equal circles of diameter d from a square of side 2d, we should pack as many full circles as possible without overlap. Regular grid packing with circle diameter equal to half the square’s side along each dimension is the natural fit.

Given Data / Assumptions:

• Square side = 2d.
• Circle diameter = d.
• No overlap; all circles must lie fully inside the square.

Concept / Approach:
Place circle centers on a 2-by-2 grid: at coordinates (±d/2, ±d/2) relative to the square’s center (or equivalently in the four quadrants), which fits exactly along both axes since two diameters equal the side 2d.

Step-by-Step Reasoning:
Across each dimension, we can fit 2 circles (since 2 * d = 2d).Total circles = 2 * 2 = 4.Attempting a fifth circle (e.g., centered) would overlap the corner circles because the distance from center to corner centers is d√2 < 2*(d/2) + (d/2), violating non-overlap.

Verification / Alternative check:
Draw the square 2d by 2d and place four circles touching each other and the square’s sides snugly—this is the standard maximal grid fit.

Why Other Options Are Wrong:
3 or 2 are suboptimal, not maximizing usage; 5 is impossible without overlap; therefore 4 is the maximal feasible count with minimum loss.

Common Pitfalls:
Thinking a center circle can fit with four corners—geometry shows the center-to-corner distances are too small to avoid overlap.

Final Answer:
4