We can rewrite the statement "$a$ and $b$ are distinct numbers" into a statement made of just variables and not-equal signs: "$a \ne b$".

Similarly, we can rewrite the statement "$a,b,$ and $c$ are distinct numbers" as "$a \ne b \ne c \ne a$".

Note that "$a \neq b, a \neq c, b \neq c$" is not valid because it is three statements, not one, and that "$(a-b)(a-c)(b-c) \neq 0$" is not valid either because it has something other than variables and not-equal signs, which in this case are the parentheses.

In the above two statements, the minimum numbers of not-equal signs $(\ne)$ used are 1 and 3, respectively.

What is the minimum number of not-equal signs used to rewrite the statement "$a,b,c,d$ are all distinct numbers" in such a way?

**Bonus:** Generalize this.

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