For what pair wise different positive integers is the value of $$\frac{a}{a + 1} + \frac{b}{b + 1} + \frac{c}{c + 1} + \frac{d}{d + 1}$$ an integer?

PLACEHOLDER ANSWER - the answer is a list of quadruples. The value is $$4 - \left(\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c + 1} + \frac{1}{d + 1}\right)$$, and four different unit fractions with denominators at least 2 add up to less than 2, so their sum must be exactly 1. The ways to write 1 as four different unit fractions are $$\{2, 3, 7, 42\}$$, $$\{2, 3, 8, 24\}$$, $$\{2, 3, 9, 18\}$$, $$\{2, 3, 10, 15\}$$, $$\{2, 4, 5, 20\}$$ and $$\{2, 4, 6, 12\}$$. So $$\{a, b, c, d\}$$ is one of $$\{1, 2, 6, 41\}$$, $$\{1, 2, 7, 23\}$$, $$\{1, 2, 8, 17\}$$, $$\{1, 2, 9, 14\}$$, $$\{1, 3, 4, 19\}$$ or $$\{1, 3, 5, 11\}$$, in any order, and the value is 3.