The trick with eeny-meeny-miney-moe is that it's long enough for us to not consciously and quickly identify whether the saying is odd or even, gives a 0, 1, or 2 on modulo 3, etc, unless we TRY to remember what it produces, or TRY to remember if it's odd or even before pointing it out. Knowing that doing so consciously ruins its capacity, we can turn to memory decay to restore some of the pseudo-random quality. basically, by sufficiently decoupling "point at A" from "choose A" to our internal cognitive algorithms...we change the way we route visual input and spit out a "point at X".

THAT"S where the randomness of eeny-meeny-miney-moe comes in...though I've probably got only one use left of it when it comes to situations with 2 items thanks to writing this up...