# Where the water is hotter

There are two perfectly identical boilers containing the same amount of water, switched on and working in identical environments. Both are setup so that the heating starts when the temperature of the water falls to exactly (or, of course, initially is below) T1 degrees and stops when it raises to exactly T2 degrees, thus keeping the temperature of the water in range T1..T2. The only available indication is whether heating is in progress.

If in an instant moment of time only one of the boilers indicates heating in progress (i.e. the other is in state of no heating), what is more probable - its water is hotter or cooler than the other's, or the chances are equal?

• Do they heat and cool linearly with time?
– Deusovi
Jun 5, 2016 at 2:29
• @Deusovi If question is about temperature changes, note the science tag, if you ask about cycle's times, it is already said that conditions are perfectly identical. Jun 5, 2016 at 2:34
• So the thermostat is set jut under $T_2$ and just over $T1$ to account for thermal inertias? Jun 5, 2016 at 4:41
• @JonathanAllan I doubt if it's matter, but may assume yes. In either case both are set and perform the same, so the temperature ranges are same. Jun 5, 2016 at 9:17
• Can T1 be set cooler than ambient temperature? Jun 5, 2016 at 14:28

More likely to be hotter

Reason:

It's about the temperature curves. On heating, the temperature rises quickly and almost linearly. The heating involves greater energy than the heat loss, otherwise the boiler would never shut off. The other boiler is cooling down from T2 to T1. The rate of heat loss is proportional to the change in temperature, that is from the current temperature between T2 and T1, and the ambient temperature. Therefore the loss is quicker at T2 than T1, meaning there is a curve, steeper at T2 than at T1, and therefore spends more time below the half-way point.

I'm going to stick with hysteresis, with the total heating time shorter than the total cooling time. With linear heating and cooling there would be 50% chance of hotter/cooler. But the non-linearity has less effect on heating than cooling. Unfortunately I'm not having the easiest time describing this.

• You should probably add pictures to describe these two curves. While this isn't done, here's how cooling and heating look like: technologytom.com/assets/images/cooling_curve102.jpg i.stack.imgur.com/fRsWY.png
– ffao
Jun 5, 2016 at 5:00
• You are on the right track. It's good to have better explanation and reasoning that takes also in account other possible factors like all possible differences between cycles' start times and heating to cooling duration ratios. Jun 5, 2016 at 9:41

The one that turns on is

Cooler. Reasoning: We have the exact same liquid, exact same environment, everything is the same except for the distance to the electrical source. When the heating element clicks off, the one closest to the source of electricity is turned off first, therefore, it starts cooling before the other. Therefore, it is the one that registers as "too cool" (like me) first, and is the first to trip the alarm to warm up again. Of course, being closer to the electrical power source, it registers itself as heating up before the other.

• I've never heard of such dependence from the distance to power source. Can you explain, please? Jun 5, 2016 at 3:04
• The amount of time that electrons travel from source to destination is regulated by distance. Which is regulated by the speed of light. speed = distance/time. Dependent, no? Jun 5, 2016 at 3:46
• I'm not sure that this logic answers the riddle, but in general, if an identical machine has longer wires leading up to it, the lower voltage it get is due to the resistance of the longer wire. The current is the same instantly despite the length of the wire, because conceptually it isn't one electron that is running around the circuit, its a sequence of electrons pushing against each other (imagine a garden hose with marbles in it - when you push a marble in one end, another pops out the other end straight away). Jun 5, 2016 at 3:54
• @Aralox not instantly, but I think we can say the time taken is negligible. However, I would say that these distances are part of what should be considered as the "same environment" and should therefore be equal. Jun 5, 2016 at 4:24
• The phrase "in an instant moment of time" negates the idea of "negligible." It's like a point on a geometry chart: Neither height nor width is "gligible." Jun 5, 2016 at 5:08