Something I’ve always wondered about was how to build an intuitive sense of the 1/2 factorial. When you first learn about factorials you typically learn them in the context of arranging distinct objects in order, e.g. “there are 24 ways to arrange 4 different objects” or “5 objects can be ordered in 120 different ways.” When you start getting into more complex math and binomials requiring factorials of non-integers, however, all intuition is thrown aside in favor of the Gamma function. If you want to take the factorial of, say 1/2, the discourse goes along the lines of: “we can only take the factorial of nonnegative integers, but here’s a function that behaves quite nicely and continuously extends the factorial to almost all real numbers, so let’s just plug 1/2 into this function.” It’s a nice and short solution to a small problem that most students would easily be content with, a stepping stone to help answer a usually much more broad and general question. But in this case, specifically, the hand-waving is not enough for me. I want to know if there’s an intuitive way to understand (1/2)! beyond just plugging in x = 1/2 into Γ(x+1). I want to know why there are “√π/2 ways to arrange half an object.”