Which of the following has the greatest mass: an egg or an egg-sized volume of deionized water?

Explanation:

You can prove it by placing the egg in sufficient volume of deionized water: if the egg floats then it will have less mass than the same volume of deionized water and if it sinks the it will have greater mass than the same volume of deionized water.

That is supported by Archimides' principle: objects immersed in a fluid (liquid deionized water in this case) will displace a volume of fluid equal to the volume of the object (the egg), and the objects immersed will feel a buoyancy force equal to the weight of the fluid displaced by the object.

So, when the object sinks is because its weight is greater than the buoyoncy force:

        ⇒ Object sinks ⇔ weight of the object > weight of displaced fluid.

Since, mass is proportional to weight, object sinks if and only if the  mass of the object is greater than the mass of displaced fluid.

When you put the egg in the the deionizes water, you will observe that it sinks.

Then, we conclude that, knowing that the egg sinks, the mass of an egg is greater than the mass of an egg-sized volume of deionized water.