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Semi-1 number (Posted on 2024-03-13) Difficulty: 3 of 5
A positive integer n is a semi-1 number if exactly half of the integers from 1 through n contain the digit 1. For example, 16 is semi-1, because exactly 8 of the integers between 1 and 16 contain the digit 1:

{1, 10, 11, 12, 13, 14, 15, 16}.

Find the largest semi-1 number you can find!

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution computer exploration plus analytic limit | Comment 2 of 4 |
ones=0;nonOnes=0;
for n=1:10000000
  ns=num2str(n);
  if contains(ns,'1')
    ones=ones+1;
  else
    nonOnes=nonOnes+1;
  end
  if ones==nonOnes
     fprintf('%13s\n',(num2sepstr(n,'%8d')))
  end
end

finds


            2
           16
           24
          160
          270
          272
        1,456
        3,398
        3,418
        3,420
        3,422
       13,120
       44,686
      118,096
      674,934
    1,062,880
     
Through 10,000,000, the stats are:

>> ones,nonOnes
ones =
     5217032
nonOnes =
     4782968

Adjusting these to 0 - 9,999,999:

ones =
     5217031
nonOnes =
     4782969
     
Then 10,000,000 - 19,999,999 all have 1's, so by the end of that span

ones =
    15217031
nonOnes =
     4782969
     
Each of the nine 10-million-number sets in the remaining 8-digit numbers have a surplus of 5217031 - 4782969 = 434062 numbers with a 1.

Going up to 100 million, there are no more semi-1 numbers, and since each 10 million has a surplus, we could have stopped at 20 million. No intervening imbalance in the opposite direction occurs between successive multiples of 10 million to temporarily balance out the ultimate excess.

That means that 1,062,880 is the absolute last semi-1 number.

  Posted by Charlie on 2024-03-13 08:34:52
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