## The shower temperature problem

Here’s an interesting problem. Imagine a large hotel in which many people are taking a shower at the same time. There isn’t enough hot water to give everyone the shower temperature they’d like and a change in temperature in one shower effects everyone else’s.

What strategy should individuals use to achieve the same temperature for everyone without large temperature changes?

That’s the question posed by Christina Matzke, an economist at the University of Bonn in Germany and a pal from Fribourg, in a curious paper on the arXiv today. They first show that there is a solution to this problem in which everyone can achieve the same shower temperature (although it may not be as hot as they’d like).

But achieving this aint’ easy. Much depends on the resolution of the taps: whether you can change the amount of hot water you getting by a small enough amount to fine tune the temperature. If you can’t, then the temperature will jump around like a cat on hot tin roof.

There are two strategies. If everyone starts with same tap settings, there is a risk of large temperature deviations as peopel change the settings in the same way. If everyone starts with different tap settings, the deviations are less but individuals are also less likely to get close to the optimal temperature.

Matzke then goes on to show that one formulation of the problem is NP-complete meaning that there ain’t a quick way of finding the solution other than testing all the combinations of tap settings.

So there ain’t a magic answer to this one. Looks like we’re destined to be unhappy showerers.

Ref: arxiv.org/abs/0801.1573: Taking a shower in Youth Hostels: risks and delights of heterogeneity

### 2 Responses to “The shower temperature problem”

1. […] on the physics arXiv blog. Related posts: • Google suggests;  • Adaptive middle agents for […]

2. […] How about a hot shower? Actually, that would feel real good after the -23 C night we had not four days ago. But what if there are too many people in the building trying to take a shower at the same time, can we all get water that’s at the same temperature? KFC at The Physics arXiv Blog explains just how (although it’s not too easy) with The shower temperature problem. […]