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Bored student (Posted on 2019-08-30) Difficulty: 5 of 5
One day an extremely bored student began writing down the positive integers in order: 1,2,3,... When she got to 20 she noticed that she had used the digit 1 twelve times (once each for 1,10,12,13,...,19 and twice for 11). She asked herself if there was some number n (greater than 1) such that in writing down the integers from 1 to n she would have used the digit 1 exactly n times.

Is there such an n?
If so, is there a largest such n?
What if the digit 1 is replaced by another digit?
What if we work in a number base other than base 10?

No Solution Yet Submitted by Danish Ahmed Khan    
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Some Thoughts partial solution | Comment 1 of 2
The first that's greater than 1 is 199981, the list of those below 10 million follows:

1   1
1   199981
1   199982
1   199983
1   199984
1   199985
1   199986
1   199987
1   199988
1   199989
1   199990
1   200000
1   200001
1   1599981
1   1599982
1   1599983
1   1599984
1   1599985
1   1599986
1   1599987
1   1599988
1   1599989
1   1599990
1   2600000
1   2600001

The left-hand 1 indicates that that is the digit used. Only 1 works within this range.

  For n = 1 To 10000000
    ns$ = LTrim(Str(n))
    For i = 1 To Len(ns)
      d = Val(Mid(ns, i, 1))
      dig(d) = dig(d) + 1
    Next
    For i = 1 To 9
      If dig(i) = n Then
         Text1.Text = Text1.Text & i & "   " & n & crlf
      End If
    Next
    DoEvents
  Next
  
These are the only ones found under 10 million. I don't see any barrier to keep the number of such things finite, but there may be such a limit.

Up to 10 million, no other digit would replace the 1. 

Altogether there seem to be 20 2-digit numbers, 300 3-digit number, 4000 4-digit numbers, 50000 5-digit numbers, etc., so the numbers keep pace.

  Posted by Charlie on 2019-08-30 13:02:05
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