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Diagnosing lack of Internet on a second router under a primary?

How do you diagnose why a second router, whose WAN port is connected to a LAN port on the primary router, doesn’t have any Internet connection?

I have a Linksys router running DD-WRT which I’ve connected as a secondary router under a primary.

If I connect to the wifi network on the primary, I have Internet.

If I connect to the wifi network of the secondary when it’s connected directly to the modem, I have Internet.

However, If I connect the second Linksys to the primary so it’s WAN port is connected to one of the primary’s LAN ports, I can connect to the second’s wifi network, but I have no wider Internet connection.

If I telnet into the secondary router and ping a Google DNS server like 8.8.8.8, it times out, so it’s not just DNS. It’s like the secondary router can’t access any IP’s from the primary.

Why would this be?

I’ve done all the usual powering off and rebooting of the modem, primary router, and secondary router, to no avail.

Both router’s have DHCP setup and I’m able to access the second’s admin interface and release/renew the DHCP, and it appears to receive new DHCP credentials from the primary.

I can also telnet into the second router and run a ping, which confirm it can’t resolve any domains.

I have no fancy configurations on either router. Both are configured to get their IPs from the upstream router, so as I understand networking, connecting one router to the LAN port of a primary should “just work” and transparently grant Internet to all devices connected to the secondary router.