Let me explain the situation.

There are 4 floors - base, 1st, 2nd, and 3rd. Each floor has a light
fixture. The previous owners used a timer to control all the lights.
To save energy my friend decided to install two 3-way light switches
at each end (base and 3rd floor) and two 4-way light switches on the
middle floors (1st and 2nd). This is an old building so all wires are
colored black and each floor also has pre-existing wiring for the
light switches but was sealed with a wall plate. We opened the wall
plate and tried to identify the wires on each floor using a non-
contact volt meter and wired the light switches but here's the
problem.

The light switches work on each floor meaning it will either turn some
lights on and some off but when you switch any switch, lights for the
base, 1st and 3rd floors will turn on and the 2nd floor light is off.
When you flip any switch again only 2nd floor light will turn on and
the others lights are off. So the lights keep working in opposite
directions. We shut the circuit breaker off and on and tried to turn
on the lights from 2nd floor and all lights turned on and off fine for
one time. But when tried to turn on the light on another floor, only
the base, 1st, and 3rd floors work and 2nd floor is off. When you
flip the switch again, 2nd floor light is turn on and the others off.

Does anybody know how to fix this wiring issue?
Thanks