MATH STINGER #84
  by Dr. Joel C. Fowler
 Associate Professor of Mathematics   

       The puzzle for this issue involves the number of digits (0 - 9) used in writing down every integer from 1 through N, where N has m digits.  For example, if N=23 then m=2 and it can be verified (by writing them all out) that 37 digits are needed to write the numbers from 1 through 23.  The puzzle is to find a short formula in terms of N and m that gives the number of digits used in general.      After solving you may want to generalize the puzzle and find a formula (using M, m, and k) for the number of "digits" used in writing all numbers from 1 through N base k where N has m digits when written base k.  Assume that k is an integer greater than 1 and that numbers base k are written with k distinct "digit" symbols. >

 

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