Can't set up Core

I got somewhat farther I think. One key thing I did was after the Core rebooted after entering wifi credentials, and started pulsing cyan (which it always did for about 1 second before switching to solid magenta) I hit enter (this refers to the “Press ENTER when your core is breathing CYAN” line in “sudo particle setup”). It went further and starting doing the rainbow flash. It still was stuck somehow so I tried a factory reset like in How to do a factory reset and started all over again and it finally seems to have added it to the build.particle.io device list. It must have had an Internet connection if only briefly.
After getting it to show up in the device list but while it was still in “shouting rainbows” mode I tried to flash the “blink-an-led” demo code on build.particle.io but that failed, with a timeout. I then tried restarting the Core and flashing the blink-an-led code again but that also failed with a timeout. Also, when the Core restarts it’s final LED resting state is solid magenta. I also tried using both the Particle and Spark Android apps on my phone and although they claim to see the Core, there is no communication–I can’t reflash Tinker, I can’t blink the led using pin D7.
Maybe it’s just a defective Core? I mean, it’s extremely effective at causing enormous frustration, but not so effective at anything else.
Any suggestions of what to try next?