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A digital circuit that literally "counts" - it progresses through a sequence of states that are representative of some value. It need not count in a natural progression, i.e., 1,2,3,4,5, etc., to be considered a counter (you can have different count sequences). Sometime it's sufficient that the counter counts to some value and then stops. Examples are Gray code, up/down counter.

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Testing a CD4018B

How can I advance the counter? I put a switch between 5V and the Clock pin but it has no effect. Resetting the counter works (I accidently advanced it by triggering a slack joint). …
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