Originally Posted by White Rabbit

Thanks Xobim. You don't need to give me a link though. It was the picture that wasn't loading, so I couldn't have seen it with or without a link.

Do you mind if I 'improve' on your map?
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Currently, we know that the left pipe is the source of the warm liquid and/or cooked spaghetti bolognese. Valves have two values: 0 and 1, 0 being its original state (the state we found it in) and 1 being an altered state (it becomes 1 when we turn the valve). So, as you can see, we have turned both valves, so their values are 1.

Now, the coloured numbers. They're very similar to your 1 and 2 system. Each pipe has a 1 and a 0, and the colour determines whether a pipe is cold or not when the valve has a specific value. To avoid confusion, boxes are used to show which pipes a particular valve affects. If both numbers have the same colour, it means that the temperature is unaffected by the valve's value, although valves from other boxes may affect the temperature.

Now, the longest pipe has a red 01, and a red/blue 01. Using my boxes and my red arrow, you can see that this is not contradictory. It just means that the pipe is cold, that the lower valve is unable to affect the pipe's temperature, and that all the bottom pipes have been deprived of their warm liquid and/or cooked spaghetti bolognese.