Every now and then I will run into potatoes that don't seem to cook, or cook really slowly. I think it has only happened in soups/stews. Is there some secret involving temperature or salt or state of mind?
What kind of potato? What size?What method of cooking?
The most common problem I've seen with cooking potatoes is when whole potatoes boiled for mashed potatoes. The problem is that they are usually added to piping hot water.
I've found that whole potatoes cook best in liquid when they are put in the pot while the water is cold. This way, the potato slowly heats up with the water and cooks through evenly. When you add potatoes to water that is already boiling hot, the outside cooks really fast and disintegrates while the center tries to catch up.
Obviously there are many other ways to cook a potato and each way will require a different answer to this question. Different varieties of potatoes have different properties and sizes which also will affect the method you use to prepare it and what dishes they work best for.
That has happened to me one before, also in a stew (about 20 years ago). It was traumatic to me and I have never gotten over it! I do not know the scientific explanation, but I remember that that particular potato had a slight tinge of green right under the skin. Since that time I have avoided using any potato with the slightest hint of green under the skin, and the problem hasn't recurred. Sorry I can't be more specific, but I hope you are assured that the problem is not you or your cooking method.
That green under the skin is solanine and it's the result of time and exposure to light and warm temperatures. They say its toxic. I've never been poisoned by it, but I also make it a point to peel off any green from my potatoes before using. the rest of the potato is perfectly good to eat. Once those eyes and green color start to appear, you just have to say good bye to the skin.
Make sure you are putting the potatoes in cold water before they go over the heat. This will help even cooking.