HAISU is a portmanteau of three Japanese words - 'hairu', to enter, 'su', number, and 'hausu', an English borrow word meaning house, of course.
Together, we get a meaning of 'enter number house', which I have roughly translated to English as 'Room Count'.
The rules are simple - draw a path from the O to the X, passing through every cell in the grid exactly once. The grid is divided into several rooms. When your path passes over a cell with the big number N, it must be the Nth time you have entered the room. If a room has a small number m in the top left corner, you must enter that room a total of m times. An example Haisu puzzle and its unique solution are shown below.
Hopefully that example puzzle clarifies the rules.
I made the puzzle below based on a fellow puzzler's logo. The shading doesn't mean anything, it's just part of the original design.
It turned out to be quite a difficult and interesting puzzle, so I've decided to post it here anyway! Enjoy :)
EDIT: The puzzle has now been changed fairly drastically past the opening steps. Apologies to those who spent extended periods of time on this question. Stackreader's and Trenin's progress still stand! It's just the later steps of the puzzle which have changed. Once again, super sorry, and this version has DEFINITELY been checked many times to make sure that it DEFINITELY works!!!