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I am doing a project at a local fair which is one of those 'steady hands' games. I am adding a twist that if you touch the wire with the loop you are dragging through it once, distractions will start.

Is it possible (using no ICs) to make a sort of trigger where when the loop touches, the trigger is triggered and the trigger output goes forever?

PeterJ
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Yes. What you are describing sounds like a flip-flop. These are fundamental building blocks of digital circuits, and are available in great quantity at very low cost as ICs. It will be more expensive, and performance will be worse, to build them from discrete components. However, it can be done:

BJT flip-flop schematic

Pulling inputs E1 or E2 low will set or reset the outputs A1 and A2. You can see that if one transistor is on, it will turn the other off, so A1 is always the complement of A2.

Phil Frost
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