I think they assume that if you’re keeping something cold you’ll have ice in it. Water undergoing a phase transition like from solid to liquid requires a good bit of energy and can essentially act as a “cold reservoir”
I see people pointing out that hot stuff is further away from room temp than cold stuff and that the difference in temp inside vs outside the cup causes the hot to become room temp faster. While this is true to some extent in that it will lose energy faster than the cold will gain, it is an exponential process and the heat transfer slows as a proportion of the temp difference. So the hot stuff will reach room temp later than cold because it started off with a larger temp difference
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