First of all, never mind all the gaping usability problems and obvious decrement in security.
The system must be storing passwords in plain text in order to make this work.
I can think of ways that don't require plain text storage (like pre-computing various partial passwords and hashing them separately), but (1) people who came up with this idea would be too stupid to have implemented that, and, (2) in practical terms, it significantly reduces the possible number of partial passwords: we can't hash all of the subsequences of the string, because there are vastly many, so we have to rather generate some reasonable number of canned subsequence positions. Such a reduction in itself is a security compromise.