Can you boil a potato whole? Without the outside turning to mush and a crunchy inside? Maybe if you leave the skin on?

Restaurants used to boil them whole with the skin on to make hash browns. It can be done.

If the temperature is low enough and the potatoes are all the same size it will work. I'm not sure if you would need to puncture the skin first to keep them from splitting.

You can easily boil whole potatoes. I would follow simon's suggestion to try to find potatoes about the same size so they cook evenly and finish about the same time. Leave the skins on. You don't have to prick the skins like you do when you bake them. The skins may split on their own while cooking though. You can easily peel them after they cool off. Take them out when they are fork tender - This is the way I cook them when I make potato salad. Works great!