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?
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.
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Posted by Charlie
on 2019-08-30 13:02:05 |
partial solution
