It's all about number theory! (1)

We can rewrite the given equations as

$(a + 1)(b + 1) = 525 = 3*5*5*7,$

$(b + 1)(c + 1) = 147 = 3*7*7$ and

$(c + 1)(d + 1) = 105 = 3*5*7.$

From the first two equations we see that $(b + 1)$ could be any of the factors common to $525$ and $147.$ These are $1, 3, 7$ and $21.$ The corresponding values for $(c + 1)$ would then be $147, 49, 21$ and $7.$

From the last two equations we see that $(c + 1)$ could be any of the factors common to $147$ and $105.$ These are $1, 3, 7$ and $21$ as well. Comparing these values to the possible values for $(c + 1)$ listed in the previous step, we see that we have reduced the possible values for $(c + 1)$ to $7$ or $21.$

Thus the possible values for $((a + 1), (b + 1), (c + 1), (d + 1))$ are $(25,21,7,15)$ and $(75,7,21,5),$ and thus $(a,b,c,d)$ could either be $(24,20,6,14)$ or $(74,6,20,4)$

However, since only $24*20*6*14 = 3*8*4*5*6*2*7 = 8!$ we find that $a - d = 24 - 14 = \boxed{10}.$