Is it true that cold water in a pan will boil faster than hot water in a pan?
No. It takes more heat to bring a pan of cold water to boiling than it does to bring an equivalent pan of hot water to boiling. You can see that this must be true by noting that the cold water must first reach the temperature of the hot water, after which both pans will be equivalent. But there are a few interesting peculiarities with freezing and boiling water. One worth noting is that water that has recently been boiled will freeze more easily than water at an equal temperature that hasn’t been boiled. That’s because boiling drives the dissolved gases out of the water so that it can crystallize more easily. The ice that forms from boiled water tends to be unusually clear because it contains very few air bubbles. Water that contains lots of dissolved air traps those air bubbles as it freezes and the air bubbles slow the freezing process.