You don't need a router. If you set the subnet mask on all of the PCs to be 255.255.0.0, then they should all be able to ping each other. Basically, opening up the subnet mask puts all of the PCs on ...
The set up is a little weird. I won't get into why, but the PCs are spread all over the house and the switch and router are not physically anywhere near each other. I have run a lot of cat 5e and cat ...