The euphoria of solving a mathematics exercise.

The first part of exercise 1.10 in 'A Course in Enumeration' by Martin Aigner asks to evaluate the sum below by counting configurations of dots.

Although the answer can be simply taken from Pascal's Triangle as

C(n+1,2)+2C(n+1,3)

the challenge of the exercise was to deduce the same answer by setting up a particular configuration of dots.

To make a long story short... that's how I spent my entire free Saturday. By not finding the answer that is. I did this ( Sunday ) morning though with great relief. I was close right away but I -sort of- forgot to finish it correctly, instead I kept searching for other, better dot configurations. In the rush of having solved another 'difficult' exercise I remembered an article with the title 'Addicted To Knowledge' which explains the feeling I had.

Aigner says in the preface: 'It is commonplace to stress the importance of exercises. To learn enumerative combinatorics one simply must do as many exercises as possible.' An exercise is only an exercise if it was a challenging exercise, I would like to add. And harder means deeper in the context of exercises.

Hint to the answer: find a dot-configuration for the sum of 1,2,3,...,n first.