How do I detect a pattern of 4 pulses from 4 different signals?
This problem fondly reminded me of the old 'watermelon box' device at the (sadly) long gone Bell Labs! Given everything you have available, here is a starting hint:



I am assuming your 4 input pulses are arriving on 4 separate lines (i.e., wires). If so, connect the top wire to the circuit as shown, i.e., where it says 'First pulse in'. Nothing is going to happen until that pulse arrives. The PNP and NPN are simply a 'homemade SCR', i.e., a device with PNPN regenerative feedback. The pulse will cause the 'SCR' to latch, so that is the memory. The latched 'SCR' turns on the upper PNP transistor, lights the LED and gives you a new power supply to use, so now you have only three remaining tasks. I hope this helps!