Peeling hard-boiled eggs can be extremely tricky if the eggs are too fresh. The number one rule, therefore, is to use eggs that are at least five days old from their packing date. The best way to do this is to first tap the eggs all over to crack the shells, then hold each egg under a slow trickle of running water as you peel the shell off, starting at the wide end. The water will flush off any bits of shell that cling on. Then back they go into cold water until completely cold. If you don't cool the eggs rapidly they will go on cooking and become overcooked, then you get the black-ring problem.
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