what you describe is a standard job interview question traditionally inflicted upon materials scientists. here is the "standard" answer:
Imagine you slice the ring into two half arcs spanning 180 degrees each. Each has radius "R".
place the two arcs back into contact, and heat them up. for each half-hoop, the radius increases to R + delta r where delta r comes from the thermal expansion.
the diameter of the pair of arcs has been increased thereby by 2*(delta r).
From this you can see that the diameter of the hole in the ring increases when the ring is heated, thereby proving what every machinist knows: to get a firm fit of a collar on a shaft, heat the collar, slide it on over the shaft, and allow to cool. the result is called a "shrink fit".