Which factors influence the moment of soft-into-hard egg during the boiling?
- size of the egg
- putting egg to cold/hot water
- colour of the egg
- atmospheric pressure (water boiling temperature)
How to accurately find the point when egg turns hard?
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Colour of the egg shell does not matter. Pigmentation is deposited, or not, depending on the breed of the bird.
There is in fact an astonishing amount of research devoted to egg cookery, as it is such an interesting topic. Whites coagulate at around 54C, yolks at about 62C. Which presents some interesting problems of course.
Here are some links that will be of interest. One is to a collection of links, often with further information in the comments.
The easiest way to achieve a specific consistency of egg is to use the Sous Vide technique. Using a water oven you can precisely control the temperature of the egg. You can also create your own water oven or other hacks.
Personally, I just bring the temperature of a normal pan of water and eggs to slightly more than my target temperature (specific amount over is based on experimentation with water levels and temperature), then turn off the range, and let it sit for ten or so minutes.
When using Sous Vide, time doesn't matter only temperature.
I know you're probably interested in what's going on inside the egg too, but if you have practical goals, you could try one of those egg "timers" that you toss in the water with the eggs. It won't compensate for differences in egg size, obviously, but hot vs. cold starting water and water boiling temperature ought to be pretty well-eliminated as variables.