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Problem of the Week 1211

Rounded Benford

Consider the positive integer powers of 2, and round each one just one significant digit. What is the most likely leading digit of the result?

Of course, one cannot compute a probability for infinitely many values as a ratio, so the question is asymptotic. For each of the 9 digits there is a limit to the proportion of times it occurs as the leading rounded digit. For which digit is this limiting probability the largest, and what is the limiting probability?

Note that the rounding decision for a number such as 85000000 does not arise, since a power of 2 is never divisible by 5.

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November 2015