How to make sure that potatoes are boiled properly?
You would probably do best to check the internal temperatures of each and every potato. You can do this with a meat thermometer.


  Potatoes are done if tender when pierced with a fork and the internal temperature reaches 210 degrees F. Use a meat thermometer to test for doneness.


[ Source: http://whatscookingamerica.net/Q-A/PotatoBaking.htm ]

(That's 98.8889 degrees Celsius, because I can't read Farenheit.)

Now, baking and boiling are different, and what you are going to end up doing with it or what type of potato you used will have an effect on this. However, you get the general idea. Thermometers will provide the best accuracy for any method of cooking.

If you want to really keep things accurate, you can use the ChefSteps method of selecting potatoes to begin with: http://www.chefsteps.com/activities/finding-perfect-french-fry-potatoes

So all your potatoes have about the same water content.

Then you can cut them all to exactly the same size.

Then you can put them in an immersion circulator/waterbath at 194°F / 90°C, use a special sous-vide bag thermometer to test the internal temperature (it pierces the bag and reseals it on the way out,) and you have the most accurate potatoes ever.

Mmmm, accurate potatoes.