Detect a happy number with constant memory

Define a process: replace a positive integer by the sum of the squares of its digits, and repeat. A number is happy if this process eventually reaches $1$; otherwise it falls into a loop that never contains $1$.

19 -> 82 -> 68 -> 100 -> 1     happy
2  -> 4  -> 16 -> 37 -> 58 -> 89 -> 145 -> 42 -> 20 -> 4 ...   loops, not happy

Decide whether n is happy in $O(1)$ extra space. A visited-set works but costs memory, the interviewer wants the constant-space version.