Visual Proofs

Each odd number wraps around the previous square as an L-shaped gnomon. 1 is a single cell. Add 3 (the L-shape) to get a 2×2 square. Add 5 to get 3×3. The pattern is self-evident: the n-th odd number always has exactly enough cells to extend the square by one row and one column.

Start with a unit square. Take half. Then half of what remains. Then half again. Each piece fills exactly half of the remaining space. The colored area approaches but never quite reaches the full square — and yet in the limit, it fills everything. Infinity does what no finite number of steps can.

Arrange the sum as a staircase. Make a copy, flip it, and fit them together. They form a perfect rectangle: n wide, (n+1) tall. The rectangle's area is n(n+1), and since it's two copies of the staircase, each staircase is n(n+1)/2. Legend says Gauss discovered this at age seven, when his teacher tried to keep him busy by summing 1 to 100. He answered instantly: 5050.

Place four identical right triangles inside a square of side (a+b). In the first arrangement, the inner space is a single square of side c — area c². Slide the triangles to the second arrangement: the inner space becomes two squares, sides a and b — area a² + b². Same four triangles, same outer square, therefore a² + b² = c². QED.