Friday, October 23, 2020

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.

Wednesday, October 07, 2020

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.

Thursday, October 01, 2020

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