I'm afraid you'll need some sort of routing table, static route, default route or routing protocol between the routers if (referring to the attached photo) PC0-5 cannot ping PC6-8.
This is because when PC0 issues a ping to an issued address (let's say 192.168.1.3) it first goes to the configured default gateway, of R2 and is then routed onto the 192.168.1.0 network. R2 is aware of all networks DIRECTLY connected to it by using ARP address resolution (something done automatically).
Because R2 is directly connected to R3 and the 10.10.10.0 network is (probably manually and static) assigned to the port, it also knows that R3 is attached. That however, is where R2's knowledge ends.
In order for PC1 to ping PC8 there MUST be a routing table record of the 192.168.20.0 network in R2. This is essentially just telling the router "If you receive a packet directed to 192.168.20.0 network, send it to R1 on the 10.10.10.0 network".
You could setup a static route in both routers to configure it all manually, but if the network is likely to change or expand then you'll want a dynamic protocol like RIP or another type.
It's been a while since I studied this, so hopefully I haven't made any mistakes. I'm sure someone can point them out if I did. Hope this helps.