Positive-integers digit-count coincidences

1

199981

199982

199983

199984

199985

199986

199987

199988

199989

199990

199991

200001

1599981

1599982

1599983

1599984

1599985

1599986

1599987

1599988

1599989

1599990

1599991

2600001

13199998

13199999

28263827

28263828

35000001

35199981

35199982

35199983

35199984

35199985

35199986

35199987

35199988

35199989

35199990

35199991

35200001

117463825

117463826

242463827

242463828

371599983

371599984

371599985

371599986

371599987

371599988

371599989

371599990

371599991

371599992

371599993

499999984

499999985

499999986

499999987

499999988

499999989

499999990

499999991

499999992

499999993

499999994

500000001

500199981

500199982

500199983

500199984

500199985

500199986

500199987

500199988

500199989

500199990

500199991

500200001

501599981

501599982

501599983

501599984

501599985

501599986

501599987

501599988

501599989

501599990

501599991

502600001

513199998

513199999

528263827

528263828

535000001

535199981

535199982

535199983

535199984

535199985

535199986

535199987

535199988

535199989

535199990

535199991

535200001

1111111110

1111111111

2222222222

3333333333

4444444444

5555555555

6666666666

7777777777

8888888888

9471736170

9471736171

9486799989

9486799990

9486799991

9486799992

9486799993

9486799994

9486799995

9486799996

9486799997

9486799998

9498399989

9498399990

9498399991

9498399992

9498399993

9498399994

9498399995

9498399996

9498399997

9498399998

9628399986

9628399987

9628399988

9628399989

9628399990

9628399991

9628399992

9628399993

9628399994

9628399995

9628399996

9757536170

9757536171

9882536171

9882536172

9971736170

9971736171

9986799989

9986799990

9986799991

9986799992

9986799993

9986799994

9986799995

9986799996

9986799997

9986799998

9998399989

9998399990

9998399991

9998399992

9998399993

9998399994

9998399995

9998399996

9998399997

9998399998

9999999999

10028263827

10028263828

10242463827

10242463828

10371599983

10371599984

10371599985

10371599986

10371599987

10371599988

10371599989

10371599990

10371599991

10371599992

10371599993

10499999984

10499999985

10499999986

10499999987

10499999988

10499999989

10499999990

10499999991

10499999992

10499999993

10499999994

10528263827

10528263828

12222222222

13333333333

14444444444

15555555555

16666666666

17777777777

18888888888

19471736170

19471736171

19486799989

19486799990

19486799991

19486799992

19486799993

19486799994

19486799995

19486799996

19486799997

19486799998

19498399989

19498399990

19498399991

19498399992

19498399993

19498399994

19498399995

19498399996

19498399997

19498399998

19628399986

19628399987

19628399988

19628399989

19628399990

19628399991

19628399992

19628399993

19628399994

19628399995

19628399996

19757536170

19757536171

19882536171

19882536172

19971736170

19971736171

19986799989

19986799990

19986799991

19986799992

19986799993

19986799994

19986799995

19986799996

19986799997

19986799998

19998399989

19998399990

19998399991

19998399992

19998399993

19998399994

19998399995

19998399996

19998399997

19998399998

19999999999

20371599983

20371599984

20371599985

20371599986

20371599987

20371599988

20371599989

20371599990

20371599991

20371599992

20371599993

20499999984

20499999985

20499999986

20499999987

20499999988

20499999989

20499999990

20499999991

20499999992

20499999993

20499999994

23333333333

24444444444

25555555555

26666666666

27777777777

28888888888

29471736170

29471736171

29486799989

29486799990

29486799991

29486799992

29486799993

29486799994

29486799995

29486799996

29486799997

29486799998

29498399989

29498399990

29498399991

29498399992

29498399993

29498399994

29498399995

29498399996

29498399997

29498399998

29628399986

29628399987

29628399988

29628399989

29628399990

29628399991

29628399992

29628399993

29628399994

29628399995

29628399996

29757536170

29757536171

29882536171

29882536172

29971736170

29971736171

29986799989

29986799990

29986799991

29986799992

29986799993

29986799994

29986799995

29986799996

29986799997

29986799998

29998399989

29998399990

29998399991

29998399992

29998399993

29998399994

29998399995

29998399996

29998399997

29998399998

29999999999

30499999984

30499999985

30499999986

30499999987

30499999988

30499999989

30499999990

30499999991

30499999992

30499999993

30499999994

34444444444

35555555555

36666666666

37777777777

38888888888

39471736170

39471736171

39486799989

39486799990

39486799991

39486799992

39486799993

39486799994

39486799995

39486799996

39486799997

39486799998

39498399989

39498399990

39498399991

39498399992

39498399993

39498399994

39498399995

39498399996

39498399997

39498399998

39628399986

39628399987

39628399988

39628399989

39628399990

39628399991

39628399992

39628399993

39628399994

39628399995

39628399996

39757536170

39757536171

39882536171

39882536172

39971736170

39971736171

39986799989

39986799990

39986799991

39986799992

39986799993

39986799994

39986799995

39986799996

39986799997

39986799998

39998399989

39998399990

39998399991

39998399992

39998399993

39998399994

39998399995

39998399996

39998399997

39998399998

39999999999

45555555555

46666666666

47777777777

48888888888

49471736170

49471736171

49486799989

49486799990

49486799991

49486799992

49486799993

49486799994

49486799995

49486799996

49486799997

49486799998

49498399989

49498399990

49498399991

49498399992

49498399993

49498399994

49498399995

49498399996

49498399997

49498399998

49628399986

49628399987

49628399988

49628399989

49628399990

49628399991

49628399992

49628399993

49628399994

49628399995

49628399996

49757536170

49757536171

49882536171

49882536172

49971736170

49971736171

49986799989

49986799990

49986799991

49986799992

49986799993

49986799994

49986799995

49986799996

49986799997

49986799998

49998399989

49998399990

49998399991

49998399992

49998399993

49998399994

49998399995

49998399996

49998399997

49998399998

49999999999

56666666666

57777777777

58888888888

59471736170

59471736171

59486799989

59486799990

59486799991

59486799992

59486799993

59486799994

59486799995

59486799996

59486799997

59486799998

59498399989

59498399990

59498399991

59498399992

59498399993

59498399994

59498399995

59498399996

59498399997

59498399998

59628399986

59628399987

59628399988

59628399989

59628399990

59628399991

59628399992

59628399993

59628399994

59628399995

59628399996

59757536170

59757536171

59882536171

59882536172

59971736170

59971736171

59986799989

59986799990

59986799991

59986799992

59986799993

59986799994

59986799995

59986799996

59986799997

59986799998

59998399989

59998399990

59998399991

59998399992

59998399993

59998399994

59998399995

59998399996

59998399997

59998399998

59999999999

67777777777

68888888888

69471736170

69471736171

69486799989

69486799990

69486799991

69486799992

69486799993

69486799994

69486799995

69486799996

69486799997

69486799998

69498399989

69498399990

69498399991

69498399992

69498399993

69498399994

69498399995

69498399996

69498399997

69498399998

69757536170

69757536171

69882536171

69882536172

69971736170

69971736171

69986799989

69986799990

69986799991

69986799992

69986799993

69986799994

69986799995

69986799996

69986799997

69986799998

69998399989

69998399990

69998399991

69998399992

69998399993

69998399994

69998399995

69998399996

69998399997

69998399998

69999999999

78888888888

79486799989

79486799990

79486799991

79486799992

79486799993

79486799994

79486799995

79486799996

79486799997

79486799998

79498399989

79498399990

79498399991

79498399992

79498399993

79498399994

79498399995

79498399996

79498399997

79498399998

79882536171

79882536172

79986799989

79986799990

79986799991

79986799992

79986799993

79986799994

79986799995

79986799996

79986799997

79986799998

79998399989

79998399990

79998399991

79998399992

79998399993

79998399994

79998399995

79998399996

79998399997

79998399998

79999999999

89999999999

100559404366

Snow Skunk

Yesterday morning, while still snowing, a skunk moved slowly along a tire track in the road.

Perhaps it had been hit or was otherwise distressed but a closeup showed nothing obvious.

Still, the skunk tracks on the road left me wondering.

Ersatz street view

I was born and grew up in Bergheim, a hillside suburb of the town of Neheim, Germany. Neheim itself came to be incorporated into the town of Arnsberg in 1975. It would be nice to be able to view the streets of Bergheim as they are now and compare them to my childhood memories.

Alas, Google's street view has no presence here, nor for that matter in most of Germany. But by a stroke of luck, a recent bicycle ride by Ronny Gängler takes us on a short stretch of Bergheimer Weg that I remember very well and, by inspection, manages to image what I believe to be my childhood house, though technically not my childhood home because that is on the opposite (not visible) part of the semi-detached structure.

Google's sidewalk path

I've been attempting to fix (in both Apple and Google maps) Weston's misplaced businesses for many years now. Because Weston has some five streets that duplicate, in whole or in part, Toronto street names, it has been relatively easy for a Toronto business that does not know its exact address or postal code to end up in Weston. The exact address and postal code can generally be found using Google search and that information can then be fed to the respective platforms. Occasionally work is also required in the pin placement of buildings, as Google's AI will often guess position on the basis of the street number and the actual site may be some distance from the street. This was the case, for instance, with Weston Lions Arena which has an address of Lawrence Ave W but is actually off Hickory Tree Rd.

Less successful has been my attempt to correct a Google Map path artifact:

The green dashed line on the south side of Denison Rd E represents a path running from the old section of Denison Rd E (which although now an adjunct of the new underpass section retains the same name) all the way to Jane St. An examination of the path in Satellite shows that it is just the south sidewalk of the street. On April 24 of this year I let Google know that the path was superfluous by first selecting it and then choosing "road does not exist" from the informant options. Google's use of "road" here is unfortunate because what I selected was the path — not the road. Likely whoever looked at my recommendation decided that Denison Rd E did in fact exist, so my suggestion has not, to date, been acted upon. I have also coloured magenta, in the above, a building outline artifact that I believe was a temporary workers' shelter while the underpass was being built but is obviously no longer there.

Google's own street view imagery archive shows how the street appeared prior to the underpass construction:

looking east, June 2009

During construction of the underpass, there were (at differing times) three paths for pedestrian traffic. The first meandered on the south side of the project from the Ethiopian Orthodox Tewahedo Church to roughly where the Life Community Church is now (but across the street):

path 1, looking west, June 2014

path 1, looking east, August 2014

The second path was what was to become the north sidewalk of the underpass:

path 2, looking east, September 2014

The third path was what was to become the south sidewalk of the underpass:

path 3 (behind the fence, left), looking west, June 2015

It is this third path that seems to have been immortalized by Google Maps. I'll point out that the east section of Denison Rd E was never under construction, so to have the path go all the way to Jane St shows that Google never really understood what was happening here.

Seasoning

October 31

November 22

A glass clearly

In The Annotated Sherlock Holmes, volume II, pages 332-333 (The Adventure of the Empty House), sidenote 15 lends credence to Sherburne Tupper Bigelow's March 1961 observation (Baker Street Journal: Those Five Volumes) on "Maybe you collect yourself, sir."

The old bookseller now mentions three titles. Then he says: "With five volumes you could just fill that gap on the second shelf." Obviously, two of the three titles must have been two-volume editions or one of the three titles must have been a three-volume edition ...

There is nothing obvious about it. Slightly earlier, the book collector was noted as having wedged under his right arm at least a dozen "precious volumes". He says: "Maybe you collect yourself, sir; here's British Birds, and Catullus, and The Holy War — a bargain every one of them. With five volumes you could just fill that gap on the second shelf."

A "volume" is a book that is part of a larger collection. In a very narrow sense, that larger collection may have the same title with individual books numbered, as with The Annotated Sherlock Holmes. However, the word was never (nor is now) always all that focused. The bookseller has at least a dozen volumes under his right arm: it's just another word for books. Connecting the five volumes to the three titles is the intervening "a bargain every one of them." The bookseller points to the three titles, then sweeps that hand over the rest of the books as he delivers the hawkers' mantra. Five volumes of this stack will fill the gap.

Proportionality

In the list of consecutive positive base-ten integers: 1, 2, 3, ...

   the 1111111111st occurrence of 1 is the first 1 in 1111111111,

   the 2222222222nd occurrence of 2 is the second 2 in 2222222222,

   the 3333333333rd occurrence of 3 is the third 3 in 3333333333,

   the 4444444444th occurrence of 4 is the fourth 4 in 4444444444,

   the 5555555555th occurrence of 5 is the fifth 5 in 5555555555,

   the 6666666666th occurrence of 6 is the sixth 6 in 6666666666,

   the 7777777777th occurrence of 7 is the seventh 7 in 7777777777,

   the 8888888888th occurrence of 8 is the eighth 8 in 8888888888, and

   the 9999999999th occurrence of 9 is the ninth 9 in 9999999999.

In other bases, the n-th occurrence of 1 as the first 1 in n is given in A023037 as a base-ten equivalent. Be careful about the offset (base two is the third term). Multiply the number by d for the n-th occurrence of d as the d-th d in n, up to d = base - 1. These n-th occurrence in n assertions are special solutions to the more general problem but they are not the only ones (all base-ten solutions are illustrated here).

On target (revisited)

one

two

three

eleven

twenty-three

twenty-four

twenty-nine

thirty-one

one hundred eight

one hundred nine

one hundred ninety-eight

one hundred ninety-nine

two hundred forty

two hundred forty-one

two hundred forty-three

two hundred forty-four

two hundred forty-five

two hundred forty-six

two hundred forty-seven

two hundred forty-eight

two hundred forty-nine

two hundred fifty

two hundred fifty-one

four hundred fifty-three

four hundred fifty-four

five hundred fifty-nine

one thousand, one hundred seventy-four

one thousand, seven hundred sixteen

five thousand, five hundred fifty-six

five thousand, five hundred fifty-seven

six thousand, nine hundred fifty-six

six thousand, nine hundred fifty-seven

fifteen thousand, seven hundred fifty-six

seventeen thousand, one hundred fifty-five

twenty-four thousand, nine hundred ninety-eight

twenty-four thousand, nine hundred ninety-nine

forty-three thousand, five hundred sixty-eight

forty-three thousand, five hundred sixty-nine

seven hundred thirty-five thousand, seven hundred fifty-nine

one million, one hundred five thousand, eight hundred five

one million, one hundred five thousand, eight hundred six

one million, one hundred five thousand, eight hundred seven

one million, one hundred seven thousand, seven hundred eighty-four

one million, one hundred seven thousand, seven hundred eighty-five

one million, five hundred eighty-four thousand, five hundred three

one million, five hundred eighty-four thousand, five hundred four

one million, seven hundred seven thousand, nine hundred forty

one million, seven hundred seven thousand, nine hundred forty-one

one million, nine hundred twenty-one thousand, five hundred sixty-six

one million, nine hundred twenty-one thousand, five hundred sixty-seven

two million, two hundred seventeen thousand, seven hundred nine

two million, two hundred seventeen thousand, seven hundred ten

two million, five hundred sixty-eight thousand, seven hundred one

two million, five hundred sixty-eight thousand, seven hundred two

three million, five hundred forty-one thousand, two hundred seventy-nine

three million, five hundred forty-one thousand, two hundred eighty

three million, five hundred forty-one thousand, two hundred eighty-one

three million, five hundred forty-one thousand, two hundred eighty-two

three million, five hundred forty-one thousand, two hundred eighty-three

three million, five hundred forty-one thousand, two hundred eighty-four

three million, five hundred forty-one thousand, two hundred eighty-five

four million, five hundred twelve thousand, five hundred ninety-six

four million, five hundred twelve thousand, five hundred ninety-seven

twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-seven

twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-eight

two hundred forty-two million, four hundred sixty-three thousand, eight hundred twenty-seven

two hundred forty-two million, four hundred sixty-three thousand, eight hundred twenty-eight

five hundred twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-seven

five hundred twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-eight

two billion, two hundred twenty-two million, two hundred twenty-two thousand, two hundred twenty-two

six billion, six hundred sixty-six million, six hundred sixty-six thousand, six hundred sixty-six

eight billion, eight hundred eighty-eight million, eight hundred eighty-eight thousand, eight hundred eighty-eight

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-one

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-two

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-three

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-four

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-five

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-six

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-seven

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-eight

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-one

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-two

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-three

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-four

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-five

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-six

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-seven

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-eight

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred eighty-six

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred eighty-seven

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred eighty-eight

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-one

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-two

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-three

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-four

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-five

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-six

nine billion, eight hundred eighty-two million, five hundred thirty-six thousand, one hundred seventy-one

nine billion, eight hundred eighty-two million, five hundred thirty-six thousand, one hundred seventy-two

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-one

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-two

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-three

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-four

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-five

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-six

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-seven

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-eight

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-one

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-two

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-three

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-four

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-five

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-six

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-seven

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-eight

ten billion, twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-seven

ten billion, twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-eight

ten billion, two hundred forty-two million, four hundred sixty-three thousand, eight hundred twenty-seven

ten billion, two hundred forty-two million, four hundred sixty-three thousand, eight hundred twenty-eight

ten billion, five hundred twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-seven

ten billion, five hundred twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-eight

twelve billion, two hundred twenty-two million, two hundred twenty-two thousand, two hundred twenty-two

sixteen billion, six hundred sixty-six million, six hundred sixty-six thousand, six hundred sixty-six

eighteen billion, eight hundred eighty-eight million, eight hundred eighty-eight thousand, eight hundred eighty-eight

...