Example URL www.[somewebsite].com/[10_digit_number]

Getting the correct number loads a page.

I know there would be 10 billion possible digits to choose from, but how long would it take? What are the resources one would need?

About a day

If we're lucky: there's no throttling, we can perform each test with a HEAD request, can perform many tests on a single HTTP connection with Keepalive, and can have many concurrent connections. 

In that case we're mostly limited by bandwidth. Say we craft a tight request that is 100 bytes, that means we need to send a total of 100 * 1010 bytes. And lets suppose we have a decent 100 Mbps connection, which will do about 10 megabytes per second. That would take 100,000 seconds - just over a day.

This is best case, in practice there are likely to be issues that prevent it working so fast. We could have multiple systems working simultaneously to make it faster - but at some point we'll overload the server.
