## Puzzle #5 – Binary-Gray counters – solution

July 5, 2007The binary-Gray puzzle from last week generated some flow of comments and emails.

Basically, the important point to notice is the amount each counter toggles while going through a complete counting cycle.

For Gray coded counter, by definition only one bit changes at a time. Therefore, for an n stage counter we get 2^n toggling events for a complete counting cycle.

For binary coded n-bit counter, we have 2^(n+1)-2 toggling events for a complete counting cycle. you could verify this by

- Taking my word for it (don’t – check it yourself)
- Writing down manually the results for a few simple cases and convince yourself it is so
- Calculate the general case, but you have to remember something about how to calculate the sum of a simple series (best way)

Anyways, given the above assumptions and the fact that per bit the Gray counter consumes 3 times more power (2 times more would also just work, but the difference would be a constant), the Gray counter will always consume more power.

3*2^n > 2^(n+1) – 2

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