The correct answer is option 1: the metal expands more than the glass when both are heated.

When a metal screw-top is stuck on a glass bottle, it is usually due to the different rates of expansion between the metal and the glass.

When both metal and glass are heated, they expand. However, metals tend to expand at a faster rate compared to glass. This difference in expansion rates can cause the metal screw-top to expand more than the glass bottle, creating a tight seal that makes it difficult to open.

To loosen the stuck screw-top, heating it gently helps to increase the temperature of both the metal and the glass. As the metal expands more than the glass, the tight seal is loosened, making it easier to unscrew the top.

Option 2 is incorrect because it states that both the metal and the glass expand identically, which is not the case. Option 3 is incorrect as it states that the metal shrinks when heated, which is contrary to the behavior of most metals when heated. Option 4 is incorrect as it suggests that both metal and glass shrink when cooled, which is not relevant to the situation described in the question.