If a circle of unit radius is divided into two parts by an arc of another circle subtending an angle 60° on the circumference of the first circle, then the radius of the arc is:

Let the first circle have centre $$O$$ and radius $$1$$. An arc of a second circle passes through two points $$A$$ and $$B$$ on the first circle, dividing it into two parts. This arc subtends an angle of $$60°$$ on the circumference of the first circle, meaning a point $$P$$ on the circumference of the first circle (on the major arc side) sees the chord $$AB$$ at an inscribed angle of $$60°$$.

By the inscribed angle theorem, the central angle $$\angle AOB = 2 \times 60° = 120°$$. The length of chord $$AB$$ is therefore $$AB = 2 \times 1 \times \sin 60° = \sqrt{3}$$.

Now let the second circle have radius $$R$$. The arc of this second circle passes through $$A$$ and $$B$$, so the chord $$AB = \sqrt{3}$$ is also a chord of the second circle. If the central angle subtended by $$AB$$ at the centre of the second circle is $$2\alpha$$, then $$\sqrt{3} = 2R \sin \alpha$$. However, without an additional constraint specifying which particular arc (i.e., the value of $$2\alpha$$ or the position of the second circle's centre), the radius $$R$$ is not uniquely determined by the given information alone — different arcs of different radii can all pass through $$A$$ and $$B$$ while dividing the unit circle into two parts.

Since the radius cannot be uniquely pinned down as $$\sqrt{3}$$, $$\frac{1}{2}$$, or $$1$$ from the given conditions, the correct answer is None of these.