Constants approximated

Goodness gracious me. XKCD’s author has come up with a list of approximations for all sorts of constants and numerical values using all kinds of unintuitive formulations. For example, he offers as a close (1 part in 25,000) approximation of Avogadro’s Constant 69π51/2. Check out the easily memorisable formula for deriving the White House Switchboard’s phone number!

How on earth did he work all these out?

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