Just looking at the problem I see that increasing the base by 2 cuts the lead digit in half, so I want a pair of numbers differing by 2 such that their squares are roughly in a 2 to 1 ratio. b=5, b+2=7 comes closest with 25/49 being very close to 1/2.

Checking: 211_b5 = 56, 110_b7 = 56. There it is: b=5 and n=56.