Here's the situation. So the other day, I'm fooling around with traceroute and ping, and I notice that the second hop for each of my is the ip address 10.10.0.1. So realizing this is an internal IP, I start poking around. So I do a ping sweep of 10.10.0.1-255, and see a bunch of hosts up. So most likely this is some router at my ISP. I'm behind a router on a 192.168.0.x network, with the router being 192.168.0.1 and the gateway.

What I am wanting to figure out is which of the 10.10.0.x addresses is my cable modem. I don't seem to ever get a hop on the cable modem. It always goes from 192.168.0.1 to 10.10.0.1. From my router's status page I see my external IP address (internet address, not a 10.10.0.x address) and my default gateway and DNS servers. For a while I was confused about why I was never getting the address my router was saying is my default gateway in my hops. Then I think I figured it out.
Code:
root@slax:~# ping -c 1 87.16.192.1 -t 1
PING 87.16.192.1 (87.16.192.1) 56(84) bytes of data.
From 192.168.0.1 icmp_seq=1 Time to live exceeded

--- 87.16.192.1 ping statistics ---
1 packets transmitted, 0 received, +1 errors, 100% packet loss, time 0ms

root@slax:~# ping -c 1 87.16.192.1 -t 2
PING 87.16.192.1 (87.16.192.1) 56(84) bytes of data.
64 bytes from 87.16.192.1: icmp_seq=1 ttl=254 time=8.20 ms

--- 87.16.192.1 ping statistics ---
1 packets transmitted, 1 received, 0% packet loss, time 0ms
rtt min/avg/max/mdev = 8.203/8.203/8.203/0.000 ms
root@slax:~# ping -c 1 10.10.0.1 -t 1
PING 10.10.0.1 (10.10.0.1) 56(84) bytes of data.
From 192.168.0.1 icmp_seq=1 Time to live exceeded

--- 10.10.0.1 ping statistics ---
1 packets transmitted, 0 received, +1 errors, 100% packet loss, time 0ms

root@slax:~# ping -c 1 10.10.0.1 -t 2
PING 10.10.0.1 (10.10.0.1) 56(84) bytes of data.
64 bytes from 10.10.0.1: icmp_seq=1 ttl=254 time=8.64 ms

--- 10.10.0.1 ping statistics ---
1 packets transmitted, 1 received, 0% packet loss, time 0ms
rtt min/avg/max/mdev = 8.648/8.648/8.648/0.000 ms
So now I'm pretty sure that 87.16.192.1 and 10.10.0.1 are different interfaces on the same device, which I think is a switch or hub or router at my ISP. But I'm still left with the question of which 10.10.0.x IP address is my cable modem. So far I could only think of one way to try and determine this. My neighbor is on the same 10.10.0.x net as me, so from his house I did two ping sweeps with my cable modem plugged in, both of which came up with the same hosts up. Then I unplugged my cable modem and did two more ping sweeps shortly after. There was no change in the hosts reported up. So that attempt at identifying my modem failed. Now I am asking you all if you can think of any way to identify myself among this network. Any ideas are appreciated. Thank you.

P.S. For those of you asking why I care about this, I'm not sure. I guess I'm just curious. Also, I changed most of the IP's, but they should still make sense as for what they represent.