Electronic – How to implement Lights-Out game using logic gates or flip flops

You have unfortuantely run into a relatively subtle problem with this particular flip-flop. (BTW, this is not metastability; that's a different problem.) It's designed to operate at high speed over a wide range of supply voltages, and one of the compromises made in its design is that it has a rather strict requirement on the clock input transition speed.

If you look at section 9 of the datasheet, the input transition rate is given as 10 ns/V maximum. This means that you need to make the clock rise or fall by 5 volts in no more than 50 ns in order for the chip to operate correctly. With an RC time constant of 10 ms, you are about 6 orders of magnitude too slow.

I think you're going into far too much detail too early on. I've designed complex state machines many years ago and this kind of problem is going to be a bit tricky to get right.

Step 1, is to get the behaviour right. Don't even think about assigning binary numbers to represent the states yet. Clearly from your postings, you haven't really got the whole behaviour worked out.

Step 2 is assign state codes and go through the an implementation technique such as the algorithimic state machine method (ASM) to design and simplify the logic, assuming you are building a logic circuit to implement it. Then you can worry about physical implementation on chips such as field programmable logic sequencers (FLPS), or even FPGA's (field programmable gate arrays), and the representation of the design can either be as a schematic (schematic capture) or a hardware description language (HDL) based.

Don't worry about any of that yet. Just focus on designing the behaviour first.

You've gone straight into a state diagram. That's your end result. That's what you're trying to achieve, so you can then build the state machine.

I think you need to start thinking about using say, a UML sequence diagram to illustrate the behaviour. Each traffic light and sensor (inductive sensor) will be an object on the sequence diagram, and the sensors will trigger a set of behaviour, in a UML sequence diagram time flows down the page, so you can easily represent time based behaviour and what happens to the lights after the inductive sensor has been triggered.

So, I'd suggest design the time based behaviour using a sequence diagram first, then when you've got that right, got what you want, then you can move to a state diagram. Then you can move to building it.