You would start to reproduce the given logic circuit in Verilog and verify its correct operation. Then add the additional logic (result counters) described in the problem text.
A structural approach seems most easy, write models for 74194 and the other logic devices, or copy it from a library. Verify the correct operation of the models if you have designed it yourself, then put it together in a design top module. Write a test bench for the game.
I appreciate this exercise problem because it treats Verilog as hardware description language rather than software.