My "L"-shaped hallway has a light switch at the end of each leg, a
switch at the intersection of the legs and a ceiling-light the middle
of each leg.

Originally, the ceiling-lights could be turned on/off from each
switch.

I've replaced the original switches with new switches but now only one
switch operates the ceiling-lights.

Should a three-way switch be installed at each end of the hallway and
the four-way switch be installed at the intersection of the legs? If
the answer is "NO", where should the various switches be installed?

How to I determine what wire gets connected to each terminal of each
switch?

You are not qualified to do this, or you would not have replaced
switches with different ones; always keep the same type.

--
Twayne

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