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Imagine you pour water and ice inside a glass cup (assume that the water and cup are at ambient temperature). Now, suppose that you get another cup but heat it up before pouring the water and ice. Which cup would cool the water faster?

I thought about this after being served water in a warm cup at a restaurant. After a moment, I immediately thought about how stupid that question was: “Clearly, the cup that was heated up would cool the water slower!” But the more I thought about it, the more confused I became.

Yes, the warm cup would transfer heat to the water, making it cool slower. However, at the same time, the heat from the cup would also reach the ice, which would make it melt faster, cooling the water faster. So, which cup would get colder faster?

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However, at the same time, the heat from the cup would also reach the ice, which would make it melt faster, cooling the water faster.

The reasoning goes awry here. The heat from the cup can reach the floating ice only through the water. In other words, the hotter cup makes the adjacent water hotter. It can't simultaneously be argued to be colder.

I do agree that the hotter cup melts the ice faster, though. Specifically, the water adjacent to the ice is at approximately 0°C. Because the water next to the hotter cup is hotter, the hotter cup increases the temperature gradient (i.e., the spatial temperature difference between the ice and the cup) and drives a larger heat flux that makes the ice melt faster. Nevertheless, the water temperature is always hotter with the hotter cup. Heat transfer can only respond to temperature gradients; it cannot "gain on" a larger gradient as if it had inertia.

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It's probably a lot more complicated than that, because water is a fluid. It will also depend on what the "ambient temperature" is.

If you put ice in a glass of water at well above $4^\circ$C, it floats at the top, melts as heat is conducted into the surface layers of the ice, and the molten ice at $0^\circ$C is denser than the warm water in the glass and drops to the bottom. This rapidly mixes cold water into the warm water, they mix turbulently, cooling the whole glass.

If the water in the glass is at $4^\circ$C it is at maximum density already, and the colder meltwater floats at the top, preventing mixing. This hugely slows the process, insulating the bulk of the water from further cooling. A pond in winter freezes slowly from the top down, and fish can survive in the water underneath.

If you stir the water with a stick, even though that is adding energy to the water, thus heating it, the stirring increases the flow of fresh warm water past the ice, speeding up the entire process. The ice melts faster, the water cools faster.

If the container is warm, the water at the edges warms. If the bulk water temperature is above $4^\circ$C then the density drops and the warm water rises, enhancing the convective circulation driven by the melting ice. If the water temperature is below $4^\circ$C the warmed water can drop, forcing a reverse circulation, pushing water up past the ice and down the sides of the glass. At some point in the middle, it might counter the flow driven by the meltwater and slow things down, but under most circumstances it has the effect of stirring the water. Warm water flows faster past the ice, melting it faster, and mixing it faster into the bulk. If the bulk is at $4^\circ$C and has stopped moving, a warm glass can start it moving again.

And yes, a warm container adds heat to the water, which counters the cooling. Which effect turns out to be bigger depends on the details. Above $4^\circ$C the warm water rises to the stop and floats there, so isn't quickly mixed into the bulk, and meltwater at $0^\circ$C still drops. Thus, it slows or even reverses the cooling of the water at the top of the glass, but has less effect on the cooling of the water in the bulk below. But the balance is likely to be complicated.

All this is probably related to the Mpemba effect. It had been observed that sometimes a tub of hot water froze faster than a tub of cold water when they were both put in the freezer. Physicists initially dismissed the claims, arguing that hot water had to cool to become cold water first before freezing, and the two processes were then the same from this point on. But they were oversimplifying; treating the block of water as if it was all at a single uniform temperature, ignoring the effects of temperature differences on convection, conduction through the different surfaces, evaporation, etc. The fact that water has a peak density at $4^\circ$C can have the peculiar effect of blocking circulation and insulating the bulk just short of freezing for a time. If the water is initially hot, it may be that it maintains enough spread in temperature to keep circulation going and drive it quickly over the $4^\circ$C peak, like a car driver taking a longer run-up to climb over a hill.

Maybe. Nobody has yet given a complete and universally accepted explanation of the Mpemba effect, and it doesn't always work. Sometimes hot water freezes faster, often it doesn't. It apparently depends on the fine details in ways we don't completely understand. So it's a very good question, and one I'd submit to experiment before trusting arguments based on thermodynamic simplifications.

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