After the last experiment this one is pretty straightforward. One 74HC32 quad OR chip, 2 timers, one SPDT switch, 2 tactile buttons, 2 LEDs, 3 10K resistors, 2 330 Ohm resistors. The wiring's the thing.
I couldn't get it to work at first, but it was just a loose connection. That, and my "breadboard-friendly" SPDT slide switch wasn't so breadboard friendly. I had to bend up the tabs on the side to get it to fit the breadboard at all, and the legs still weren't long enough. But it works.
Anyway, the switch is for Art Fleming to activate and deactivate the contestants buttons. The buttons are tied to Vcc through a pull-up resistor. A jumper from one side of the switch to the side of button 1 1 not connected to Vcc and then a second jumper from there to the corresponding button on button 2. The other side of B1 goes to OR Gate1, input 1. The same for B2, the corresponding side goes to OR Gate 2, input1. Both input2s are connected to Gate3 output. Gate3 inputs are connected to the output pins of the 555s.
So, in order for the output of either 555 to go H, the input on trigger pin 2 must be L. The trigger pins are tied to the outputs of OR Gates 1 and 2. OR output is H if either input is H. One input is tied to 555 output, which is L until triggered by it's corresponding OR Gate. Button outputs are H if no action is taken, so the 555s are not triggered. When a button is pushed, the voltage from the button goes negative, making both inputs L, and the corresponding output L, thus triggering the corresponding 555 output. Both 555 outputs are connected to LEDs, so the LED lights and stays on until reset. Once one 555 output is H, Gate 3 output is H, and neither button has any effect because the corresponding outputs will be H.
When Art Fleming activates the contestants buttons and asks a question, the first contestant to press the button lights his/her LED and locks the other contestant out. The other side of the slide switch is connected to the 555 reset pins (4), so when Art slides back the LED turns off. When he's ready for the next question, he flips the switch back and the cycle repeats.
Here's the video.