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In Peter’s home bathroom, there was a standard shower stall with everything working in a conventional fashion. Then Peter used some simple plumbing tools, and some ordinary, low-cost plumbing parts to make a modification. The next day, Peter took a shower first while Pam waited her turn. Peter finished and said, “Enjoy the shower, Pam!” She found both shower faucets in the off position. Then she turned on only the hot water faucet, and the water coming out the shower head alternated every 1 second, from hot to cold to hot. After 3 seconds, she shut off the hot water faucet and turned on only the cold water faucet, and the water coming out the shower head alternated every 1 second, from cold to hot to cold. After 3 seconds, she turned off the cold water faucet and shouted out, “Peter, what did you do?!”

What did Peter do to cause the water temperature to alternate?

Peter made no changes to the tank water heater, the faucets, or the shower head, which are all still working in a conventional fashion. Water still flows one-way only through all of the pipes. The only means of providing hot water to the shower head is from the tank water heater. The only means for providing cold water to the shower head is from the water main (or from a depleted tank water heater or while waiting for the hot water to reach the shower head). The supply from the water main is of constant temperature and pressure. Nobody but Pam touched the plumbing system after Peter told her to enjoy the shower.

Update / Hint

Draw a simplified plumbing schematic of a standard shower stall, and the solution should be easier to find. Below is such a drawing.

simplified plumbing schematic of a standard shower stall

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  • $\begingroup$ The washing machine started? Or maybe the sprinkler system? $\endgroup$ Jan 27 at 22:24

3 Answers 3

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I can see an answer but it means it is not an accident, Peter has a weird sense of humor.

Note:

First note that the description "alternated every second from hot to cold to hot" is a bit misleading, since the water flows only for 3 seconds. In fact the water is hot with a 1 second intrusion of cold water.

What Peter did:

Peter installed a heat-exchanger between the hot and cold water pipes leading to the faucets. He closed the cold faucet last. That cooled down the water in the heat-exchanger, including the hot side.

When Pam opened the hot water, some hot water came from the pipe, then the cold water from the heat-exchanger, then hot water again. That action also heated the water from the cold side in the heat-exchanger. So when Pam opened the cold water, the reverse thing happened. She got cold water from the pipe, then hot water that was heated in the heat-exchanger, then the cold water again.

The obvious solution to the problem:

Pam should divorce or find a new boyfriend.

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  • $\begingroup$ It's a creative answer, but it goes against the conditions that "The only means of providing hot water to the shower head is from the tank water heater. The only means for providing cold water to the shower head is from the water main" and it might exceed the "low-cost plumbing parts" that Peter used. $\endgroup$
    – FlanMan
    Jan 28 at 23:36
  • $\begingroup$ I don't agree with your first objection. The water flows from the water heater to the hot faucet and from the water mains to the cold faucet. It is just going thru the heat exchanger. I agree to the 2nd objection. $\endgroup$
    – Florian F
    Jan 29 at 10:20
  • $\begingroup$ Why divorce? I think a good plumber earns a lot. $\endgroup$
    – WhatsUp
    Jan 31 at 1:45
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To find a valid answer, start by modifying the simplified plumbing schematic as follows:

Do not modify the simplified plumbing schematic, since the answer can use essentially the same diagram. But note that the diagram is "not to scale". Can you find the rest of the answer before reading further?

Peter made the following physical modifications:

Peter made the pipe longer, from the junction near the faucets, to the shower head. This is the gray section of pipe in the diagram shown in the OP. He used PVC pipe, which has low thermal conductivity.

At the end of his shower, Peter did the following:

Peter alternately opened and closed the hot and cold faucets, leaving alternating segments of hot and cold water in the pipe to the shower head.

This left the water in the following state:

simplified plumbing schematic of a standard shower stall with segmented water temperatures

So when Pam ran the water in the shower:

The water came out hot, cold, hot, cold, hot and cold. It didn’t matter which faucet(s) she opened in which order, and after 6 seconds of running water, the water would then be the temperature of whichever faucet(s) she opened last.

This puzzle was inspired by:

A real shower that I took. At one point, I ran only cold water, then I shut off the shower for a minute, and then I turned on only the hot faucet. Cold water came out for a second, followed by hot water.

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Florian F's answer gets part of the solution but doesn't realy explain the cold/hot/cold nature of the water flow. So, at the cost of wasting water , how about tthe following:

There are four inefficient heat exchangers, two with hot supply and two with cold supply - these operate continuously, with water excaping to the drain. The hot supply has a pipe running through a cold heat exchanger A followed by a hot heat exchanger B followed by the hot tap. If the system has been first allowed to reach thermal equilibrium then when the hot tap is turned on, the first flow is hot - (heated by exchanger B), then there is a burst of cold water (cooled by exchanger A), then hot water (heated by the water heater. As the heat exchangers are inefficient, the cold burst can get through exchanger B without heating significantly.

There is an equal and opposite effect when operating the cold tap.

Note that if the taps were left on for more than 3 seconds there would be no more rapid temperature exchanges - so hot tap would remain hot and cold tap would remain cold. Also if Pam repeated the 'experiment' without waiting for long enough for the inefficient heat exchangers to heat/cool the water in the inside pipe, then the temperature change wouldn't be observed.

Also note that the diagram is only an approximation of the required pipe lengths - to avoid an initial splash of room-temperature water, the section of pipe between Hx B and the hot tap, and from the hot-tap to the shower nozzle should be very short (and similar for Hx D and the cold tap route). If these pipes are not suitably short then water flow when turning on hot would be [room-temp/hot/cold/hot].

enter image description here

For a 'cheap and inefficient' heat exchanger that can be made from "some simple plumbing tools, and some ordinary, low-cost plumbing parts" how about the following? Length of the heat exchanger would be ~a few 10's of cm - approximating the length that one second of normal water flow runs a long in one second. enter image description here

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  • $\begingroup$ I don't see how a heat exchanger would change the water temperature in 3 seconds from hot to cold to hot, or from cold to hot to cold. I think it would more slowly change the temperatures in the hot and cold chambers to warm in both chambers after it reaches equilibrium. $\endgroup$
    – FlanMan
    Feb 1 at 14:07
  • $\begingroup$ @FlanMan See updates to answer your question $\endgroup$
    – Penguino
    Feb 1 at 22:07

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