A circle of radius 1 has a diameter of 2. We can arrange 4 circles in a 2x2 grid pattern, each touching its neighbors, to cover the square. Each circle covers a 2x2 portion, and thus 4 circles will cover the entire 4x4 square without leaving any uncovered areas.

["How Four Unit Circles Fit Perfectly in a 4x4 Square: A Strat simple Geometric Solution", "When it comes to clever geometry arrangements, few examples illustrate symmetry and tiling efficiency as beautifully as arranging four unit-radius circles in a 2x2 grid pattern within a 4×4 square. This configuration leverages the fundamental relationship between a circle’s radius and its diameter—specifically, that a circle with radius 1 has a diameter of exactly 2—and transforms a simple mathematical fact into a visually compelling and functionally precise layout.", "### The Core Concept: Diameter and Diameter-Aligned Coverage", "At the heart of this arrangement is a clear geometric truth: the diameter of a circle with radius 1 is 2. This means the full width across the circle—from one end of the horizon to the opposite—is exactly 2 units. When circles are arranged so that each touches its neighbors horizontally and vertically, their combined linear span perfectly fills a segment of 2 units, enabling tight, non-overlapping placement across a 4-unit side length — the size of the large square.", "### Arranging Circles in a 2x2 Grid Pattern", "By organizing the four unit circles in a 2-by-2 grid, each circle occupies a quadrant of the larger square. Consider dividing the 4×4 square into four equal 2×2 sections — each serving as the "home" for one circle. Within each quadrant, a circle of radius 1 fits snugly, touching its adjacent circles but with no gaps, since their radii extend exactly across the dividing line. The horizontal and vertical tangency ensures no uncovered spots remain — the entire square is fully covered.", "### Visualizing the Coverage", "Imagine dividing the 4×4 square into four quadrants, each 2 units wide and 2 units tall. At the center of each quadrant, center each circle so its center lies 1 unit from each edge of the square. Then, with radius 1, each circle extends fully across the quadrant, touching its neighbor circles at the midpoints of shared edges. This symmetric layout ensures full coverage without overlap or sparsity.", "### Mathematical Confirmation", "- Circle diameter = 2×radius = 2×1 = 2\n- Four circles aligned horizontally and vertically occupy segments of size 2 across, so:\n - Horizontally: 2 units per row, with 4 rows → total width 4\n - Vertically: 2 units per column, with 4 columns → total height 4\n- The entire square (4×4) is thus precisely filled by the four circles, confirming perfect tiling with no gaps or omissions.", "### Why This Geometric Arrangement Matters", "Beyond being a neat puzzle, this configuration demonstrates key principles in tessellation and efficient space use. By respecting the relationships between a circle’s diameter and the covering dimensions, this method maximizes coverage and minimizes wasted space—an approach relevant in design, manufacturing, and resource optimization.", "### Applications and Connectivity", "Such patterns inspire modern grap..\n- architectural design\n- grid-based terrain planning\n- packing optimization algorithms\n- educational tools for teaching geometry", "Moreover, this 2D packing concept scales to three dimensions, influencing container design, logistics, and even cellular network coverage planning where coverage zones resemble spherical regions.", "---", "Conclusion", "Arranging four unit-radius circles in a 2×2 grid spread across a 4×4 square exemplifies the elegance of geometric compatibility. With each circle’s diameter of 2 perfectly matching the segment sizes, the layout achieves full, seamless coverage. Understanding these spatial relationships not only deepens appreciation for Euclidean simplicity but also illuminates practical methods for efficient spatial design. Whether for math enthusiasts, engineers, or educators, this layout serves as a powerful illustration of how fundamental mathematics meets functional real-world utility."]