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96 Tees (Posted on 2023-11-01) Difficulty: 3 of 5
A "Tee" is a decimal digit that includes the letter "t", or simply "two, three or eight".

What is the smallest positive integer N such that CONCAT(N^1,N^2, ... ,N^7,N^8), i.e. the concatenation of the first 8 powers of N includes 96 "Tees".

No Solution Yet Submitted by Larry    
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Solution computer solution | Comment 1 of 2
clearvars
for n= sym(50): sym(3000000)
  c=[char(string(n)) ...
    char(string(n^2)) ...
    char(string(n^3)) ...
    char(string(n^4)) ...
    char(string(n^5)) ...
    char(string(n^6)) ...
    char(string(n^7)) ...
    char(string(n^8)) ...
  ];
  ct=length(find(c=='2'|c=='3'|c=='8'));
  if ct==96
    disp([n ct])
  end
end

finds first (and once found, I stopped the program):

[1186022, 96]

so the answer is 1,186,022.

The powers are

1186022, 
1406648184484, 
1668315693058082648, 
1978659114912133298346256, 
2346733240786318158771223233632, 
2783277251703870635302163721998691904, 
3301028052620328058622342821892332569365888, 
3915091893024866724743388278306388058584469217536

The concatenation is 224 digits long.

In the vicinity of 1186022, the 96 occurrences is a high outlier. It's the only one that's 96 between 1186000 and 1187000. The first 50 counts in this area are


[1186000, 28]
[1186001, 69]
[1186002, 63]
[1186003, 77]
[1186004, 70]
[1186005, 71]
[1186006, 68]
[1186007, 66]
[1186008, 71]
[1186009, 67]
[1186010, 52]
[1186011, 61]
[1186012, 80]
[1186013, 59]
[1186014, 63]
[1186015, 75]
[1186016, 69]
[1186017, 62]
[1186018, 77]
[1186019, 51]
[1186020, 67]
[1186021, 67]
[1186022, 96] <-- the only 96
[1186023, 77]
[1186024, 64]
[1186025, 73]
[1186026, 69]
[1186027, 71]
[1186028, 62]
[1186029, 67]
[1186030, 73]
[1186031, 72]
[1186032, 78]
[1186033, 80]
[1186034, 66]
[1186035, 67]
[1186036, 66]
[1186037, 80]
[1186038, 73]
[1186039, 74]
[1186040, 55]
[1186041, 74]
[1186042, 67]
[1186043, 59]
[1186044, 62]
[1186045, 67]
[1186046, 68]
[1186047, 65]
[1186048, 66]
[1186049, 58]
[1186050, 62]

in keeping with what we might expect as 224 * .3 = 67.2

In contrast a different range of one thousand, higer up by an order of magnitude:

[21002023, 96]
[21002152, 96]
[21002221, 96]
[21002467, 96]
[21002823, 96]
[21002935, 96]

where the lengths are of the order of 267, and 267 * .3 = 80.1.

or

[31002131, 96]
[31002206, 96]
[31002207, 96]
[31002233, 96]
[31002265, 96]
[31002428, 96]
[31002532, 96]
[31002536, 96]
[31002631, 96]
[31002688, 96]
[31002787, 96]
[31002808, 96]
[31002853, 96]
[31002911, 96]
[31002984, 96]

where the concatenations are closer to 272 long and 272 * .3 = 81.6.

or to this span of 1000:

[331002031, 96]
[331002033, 96]
[331002047, 96]
[331002049, 96]
[331002056, 96]
[331002061, 96]
[331002064, 96]
[331002081, 96]
[331002120, 96]
[331002121, 96]
[331002225, 96]
[331002262, 96]
[331002338, 96]
[331002363, 96]
[331002424, 96]
[331002468, 96]
[331002536, 96]
[331002537, 96]
[331002552, 96]
[331002625, 96]
[331002628, 96]
[331002669, 96]
[331002724, 96]
[331002756, 96]
[331002763, 96]
[331002782, 96]
[331002802, 96]
[331002807, 96]
[331002814, 96]
[331002839, 96]
[331002852, 96]
[331002867, 96]
[331002896, 96]

where the concatenated lengths are over 310.

I'm sure that as the numbers get higher, eventually 96 will become an outlier as lower than the average altogether.

  Posted by Charlie on 2023-11-01 15:52:56
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