A number that provably contains all finite bit patterns is X = . 1 10 11 100 101 110 111 0000 0001 ... (the binary version of Champernowne's constant [1])

A number that provably contains all finite bit patterns is X = . 1 10 11 100 101 110 111 0000 0001 ... (the binary version of Champernowne's constant [1]) but the index at which any pattern occurs takes more bits to write than that pattern itself, and the same would hold (on average) for patterns in pi, assuming it actually is normal. [1] https://en.wikipedia.org/wiki/Champernowne_constant https://en.wikipedia.org/wiki/Champernowne_constant