You are right that one number only will have a Uniform distribution (or what you call "even"), however, it will be a discrete uniform distribution (if I understand correctly that the selected number is an integer) and therefore not a straight line parallel to the X-axis but a series of 16 dots parallel to the X-axis. Also - I don't think that the sum of 3 uniform distribution is close enough to the bell curve (the Normal or Gaussian distribution). I am not sure how many you need, but probably at least 10.

You are right that one number only will have a Uniform distribution (or what you call "even"), however, it will be a discrete uniform distribution (if I understand correctly that the selected number is an integer) and therefore not a straight line parallel to the X-axis but a series of 16 dots parallel to the X-axis. Also - I don't think that the sum of 3 uniform distributions is close enough to the bell curve (the Normal or Gaussian distribution). I am not sure how many you need, but probably at least 10. Having read your question again, I believe you don't want the score to have an exact "bell curve" distribution but just a non-uniform distribution. If this is your question, then you are right - the distribution of the sum (or average) of 2 or 3 or more scores becomes a triangular distribution, which means smaller probabilities on both ends and higher probabilities towards the center.