How Many Threes?Date: 06/12/99 at 18:35:17 From: ken logan Subject: Problem Solving (Graduate Level) Dear Dr. Math, If all the numbers from 1 to 333,333 were written out, how many times would the digit 3 be used? Date: 06/13/99 at 08:07:50 From: Doctor Anthony Subject: Re: Problem Solving (Graduate Level) Consider the righthand column alone starting at 000000. 000000 000001 000002 ...... 333329 <- there are 333330 rows to here 333330 333331 333332 333333 In the first column up to row 333330 there will be equal numbers of all digits 0 to 9, so the number of 3's is 1/10th of this = 33333. We then add the final 3 in this column, giving 33334 altogether. Considering just the second column we have 000000 000001 ...... 333299 <- 333300 rows to here 333300 ...... 333310 ...... 333320 ...... 333330 333331 333332 333333 Considering the second column alone there are 1/10 * 333300 + 4 = 33334 threes, which is the same result as for the first column. Similarly the other columns each give 33334 lots of 3's. The total number of 3's is 6 x 33334 = 200004 - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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