Do routers have to be in a separate subnet? [feedly]

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Do routers have to be in a separate subnet?
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So I'm training for a test on my Cisco course where we'll have an assignment to make some subnetting and came to wonder: do two routers have to be in a separate (in respect to all the other computers in the network) subnet to allow proper communication between clients on one end with the clients from the other?

An example assignment we were given was to create four subnets capable of housing 5, 4, 3 and 1 (administrator) computers. So I did something like this:

Sub1: 10.0.0.0/29
Sub2 10.0.0.8/29
Sub3: 10.0.0.16/29
Sub4: 10.0.0.24/30

So we have something like this:

Now, in this example, to allow proepr communication between all the PCs, I put Router0 and Router1 into a separate subnet so that they're 10.0.0.29/30 and .30/30 respectively on their Serial ports. Is this a correct way of doing this or should such task be done in such a manner that the two routers wouldn't need a separate network?

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