The fact that near boiling water cools down quicker than warm water used to be a well-known kitchen knowledge bit. Like my grandma who wasn't a physicist at all knew it. I guess in some places (particularly those where people microwave water) that part of culture is lost cause there's at least a whole generation which hasn't done cooking.

nice benchmark! coffee-based Turing test.

Is this for real?

This is like someone with no background in physics or engineering wondering "can a LLM predict the trajectory of my golf ball". They then pontificate about how absolutely complex all of the interacting phenomenon must be! What if there was wind? I didn't tell it what elevation I was at! How could it know the air density!? What if the golf ball wasn't a perfect sphere!!? O M G

And then being amazed when it gets the generic shape of a ballistic curve subject to air resistance.

This speaks far more to the ignorance of the author than something mind boggling about the LLM.

That initial drop reminds me of one of the things that stuck to me from my thermodynamic lectures / tests: If you want to drink coffee at a drinkable temperature in t=15min, will it be colder if you add the milk first or wait 15min and then add milk? (=waiting 15 min because the temperature differential is greater and causes a larger drop). Almost useless fact, but it always comes up when making coffee.

On a related note, I have been working on an app that helps determine the correct grinder setting when dialing in espresso. After logging two shots with the same setup (grinder, coffeee machine, basket etc), it then uses machine learning (and some other stuff that I am still improving) to predict the correct setting for your grinder based on the machine temperature, the weight of the shot etc.

https://apps.apple.com/ph/app/grind-finer-app/id6760079211

Its far from perfect when it comes to predictions right now but I expect to have massive improvements over the coming weeks. For now it works ok as an espresso log at least.

I'm hoping after a few tweaks I can save people a lot of wasted coffee!

  Does that seem hard? I think it’s hard. The relevant physical phenomena include at least..,

In most engineering problems, the starting point is recognizing that usually one or two key things will dominate and the rest won’t matter.

There's a simple differential equation often taught in intro calc courses, "Newton's Law of Cooling/Heating," which basically says that the rate of heat loss is proportional to the difference in temperature between a substance and its environment. I'm curious what that'd look like here. It's a very simple model, of course, not taking into account all the variables that Dynomight points out, but if a simple model can be nearly as predictive as more complex models...

I'm also curious to see the details of the models that Dynomight's LLMs produced!

  LLM  T(t)  Cost
  Kimi K2.5 (reasoning)  20 + 52.9 exp(-t/3600)+ 27.1 exp(-t/80)  $0.01
  Gemini 3.1 Pro  20 + 53 exp(-t/2500) + 27 exp(-t/149.25)  $0.09
  GPT 5.4  20 + 54.6 exp(-t/2920) + 25.4 exp(-t/68.1)  $0.11
  Claude 4.6 Opus (reasoning)  20 + 55 exp(-t/1700) + 25 exp(-t/43)  $0.61 (eeek)
  Qwen3-235B  20 + 53.17 exp(-t/1414.43)  $0.009
  GLM-4.7 (reasoning)  20 + 53.2 exp(-t/2500)  $0.03

Irrelevant to your specific cup of coffee its giving you a generic answer.

The problem is both highly complex, but fairly easy to model. Engineers have been doing this for over a century.

Of all the cooling modes identified by the author, one will dominate. And it is almost certainly going to have an exponential relationship with time.

Once this mode decays below the next fastest will this new fastest mode will dominate.

All the LLM has to do, then, is give a reasonable estimate for the Q for:

$T = To exp(-Qt)$

This is not too hard to fit if your training set has the internet within itself.

I would have been more interested to see the equations than the plots, but I would have been most interested to see the plots in log space. There, each cooling mode is a straight line.

The data collected, btw, appears to have at least two exponential modes within it.

[The author did not list the temperature dependance of heat capacity, which for pure water is fairly constant]

" Does that seem hard? I think it’s hard. The relevant physical phenomena include at least"

Imo no, this seems like something that would be in multiple scientific papers so a LLM would be able to generate the answer based on predictive text.

It isn't that surprising that it works well, this problem is fairly well known and some simple heat equations would lead to the result, about which there is a lot of training data online.

The water temperature drops quickly because the room temperature ceramic mug is getting heated to near equilibrium with the water. If you used a vacuum sealed mug(thermos) then the water temp would drop a bit but not much at all initially.

... and so another benchmark is born.