Find the number of $6$-tuples $(a_1,a_2, a_3,a_4, a_5,a_6)$ of distinct positive integers satisfying the following two conditions: (a) $a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 30$ (b) We can write $a_1,a_2, a_3,a_4, a_5,a_6$ on sides of a hexagon such that after a finite number of time choosing a vertex of the hexagon and adding $1$ to the two numbers written on two sides adjacent to the vertex, we obtain a hexagon with equal numbers on its sides. Lê Anh Vinh