Something I noticed last month:
10 in base ten…
is 101 in base three…
and 1010 in base two.
Notice that the representation changed by just adding an extra digit. Also, this is the maximum number of times that this could happen for 10; base two is the smallest base, while a two-digit representation is the shortest that could have a pattern like this (since 10 is represented as A for any base higher than ten).
Naturally, I’m curious if there’s another number like this. 64 comes close:
10 in base sixty-four…
100 in base eight…
1000 in base four…
1 000 000 in base two.
But the pattern breaks for base three.
Mean Green Math 