The usual rounded rectangle glues a quarter circle onto a straight edge. The edge has zero curvature, the arc has a lot, and the change happens in a single point. A smoother curve, a superellipse, bends in without that jump. Eight rounds: spot the smooth one. Chance alone gives 4 of 8.
Both shapes have the same width, height and area. Corners get tighter each round, so the seam gets harder to see.
Matching superellipse exponent n =
Largest gap between outlines: of the width
Curvature jump at the seam: (in 1 / half-width) at one point.
The arc corner steps from 0 to a constant and back. The superellipse rises and falls with no step. The gap between the two outlines is tiny, yet your eye reads the step.
All shapes and numbers on this page are computed live in your browser from the equations below; no external data set is used. Superellipse |x|n + |y|n = 1 (Lame curve): Wikipedia, Superellipse (secondary source; read for the definition only). Curvature: Wikipedia, Curvature. Curvature of the implicit curve is evaluated with the standard formula for F(x,y)=0. The area-matched exponent n is found by bisection on the polygon area of each outline. The 4 of 8 chance level is simple arithmetic (8 rounds, 50 percent each). I did not test how many readers actually pass.