What is the most frequent digit among the numbers between 1 and 1000 (inclusive)? Consider only base-10 whole numbers.

#### Solution

1

No need to actually tally them up. We can deduce that generally all digits should have the same frequency, because, for example, for every 1 in the ones place (1, 11, 21, etc.) there should be a corresponding number that has a 2 in the ones place (2, 12, 22, etc.). The same is true for the other digits. Furthermore, the same logic applies to the tens place and hundreds place.

However, it is not true for 0, because there is no 01, 02, 03, etc., resulting in much fewer instances of 0.

Finally, since the question ends at 1000, that adds one additional 1 and three additional 0’s, with no corresponding number that includes the other digits. Since there were much fewer 0’s, this does not change the fact that 0 is the least frequent digit, but it does give 1 one more instance than the other digits. Therefore the most frequent digit between 1 and 1000 is 1.