SUCCESS!
At your suggestion, I attempted to update the driver of the ATI video card in one machine. Not surprisingly, "Update driver" says my 2001 version was the newest available. Went to the ATI site with device description and found another from 2002. Upgraded. Presto! It works.
Microsoft hasn't updated their XP driver base since 2001?
How sad...
(Rule of thumb: NEVER allow Windows Update to update ANY drivers; in my experience, 80% of the time it either doesn't work, or it breaks things.)
I guess it applies to "Driver update" in the control panel also.
Thanks again. I'll try it on the other 2 and see where we go.
Clouseau