Preliminary analysis: one in every three numbers has this undesirable property; the edges mess this up but shouldn't be able to add more than two extra undesirables; it should be impossible to have more than ceil(33.333) + 2 = 36 undesirables.

(Also, since six bits will serve to identify 64 different numbers, it should be impossible to have more than 36 numbers that can't be identified that way.)

I'll update with boring manual data later.

---- update ----

That was wrong; using a naive guessing method, I found 37 values that require 7 guesses.

Thanks for spotting it! Exactly right, I fixed the text.