Is it possible to change the internal local ip address to any other one?

Question

linux

Given:
2 computers on the network connected by a local network using a router.
One computer has IP: 192.168.0.3, the other has 192.168.0.4.
Task:
Change the IP of one computer to 12.45.78.10, the other to IP to 210.134.88.4. At the same time, so that they can connect to each other without using the Internet. There are only 2 PCs and a router between them.
Question:
Is it possible at all and approximately how can it be implemented?
PS: the question has no practical value behind it, just an interesting argument.

Answer 1

Let's start with the fact that the address 12.45.78.0 is wrong, ".0" is always a network address, not a specific host. Addresses of specific hosts cannot be ".0" (this is with any subnet mask, with masks less than 255.255.255.0 there are other exceptions).
And now the answer, if you correct the above:
There are two options for implementing the task:
1. Place them on the same subnet, this will be the mask 0.0.0.0. Then you don't need a router. Forget about interacting with the rest of the Internet, with this mask ALL hosts with this mask are on your local network.
2. Take a router with support for multiple addresses on Ethernet (not every router with stock firmware can do this) and assign addresses to Ethernet (which is LAN) from the same subnets as hosts (computers). Accordingly, do not forget the default routing on the hosts to the corresponding addresses of the router. And the router itself will figure out where to send the packets, since both networks are on its Ethernet-e.