Underline, reproduce
Dall.e creation
If we underline the... « first, second, third, fourth, fifth, sixth, seventh, eighth, ninth, tenth, eleventh,… » letter of this sentence S1, we see that the successive underlined letters simply reproduce S1. No big deal.
If we decide not to start S1 with "first" but with "sixth", we can reproduce the new S6 quite easily:
S6 = sixth, second, third, fourth, fifth, first, seventh, eighth, ninth, tenth, eleventh,…
S6 was reproduced in a slightly less mechanical manner than S1: just two words of S1 swapped here ("first" and "sixth", in red, above).
If we start S8 with "eighth" we just swap again a pair of words of S1 ("first" and "eighth", in red, hereunder):
S8 = eighth, second, third, fourth, fifth, sixth, seventh, first, ninth, tenth, eleventh,…
What about starting S9 with "ninth"? How many swaps do we need to reproduce S9? This is more tricky! (Indeed, we couldn't compute the first 100 terms so far):
S9 = ninth, twentieth, thirtieth, fourth, ninetieth, sixth, seventh, eighth,…
Questions
how many terms of S1 must "swap" to produce S9?
And S10?
The same "technique" can be used with digits, of course. The sequence D1 is the equivalent of the sequence S1:
D1 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ...
But if we start with a(1) = 2, we get (hopefully) for D2:
D2 = 2, 20, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 22, 31, 1, 14, 18, 16, 41, 21, 15, 17, 23, 42, 44, 100, ...
What about, say, D3 to D9?
__________________
Next day update
Giorgos Kalogeropoulos was quick to send the hereunder terms:
D2
2, 20, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 22, 31, 1, 14, 18, 16, 41, 21, 15, 17, 23, 42, 44, 100, 24, 29, 25, 26, 19, 28, 33, 35, 34, 37, 32, 39, 30, 38, 40, 43, 36, 45, 46, 47, 48, 49, 50, 51, 52, 53, 180, 55, 56, 57, 27, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 58, 75, 76, 77, 78, 79, 54, 81, 82, 83, 80, 85, 86, 87, 74, 89, 90, 91, 92, 93, 600, 95, 96, 97, 98, 94, 88
D3
3, 2, 30, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1, 19, 32, 40, 22, 21, 24, 100, 26, 23, 25, 29, 18, 31, 20, 33, 34, 35, 36, 27, 38, 39, 28, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 37, 52, 53, 51, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 54, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 74
D4
4, 2, 3, 40, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1, 21, 22, 23, 24, 41, 26, 25, 28, 27, 30, 29, 32, 31, 34, 35, 36, 37, 38, 20, 39, 33, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100
D5
5, 2, 3, 4, 50, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 1, 23, 24, 25, 26, 27, 28, 29, 52, 90, 32, 31, 34, 35, 33, 37, 36, 39, 38, 22, 40, 43, 42, 45, 44, 47, 46, 30, 200, 48, 49, 53, 54, 51, 56, 57, 58, 59, 60, 41, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 55, 72, 74, 75, 76, 77, 78, 73, 80, 61, 82, 83, 84, 85, 86, 87, 88, 89, 79, 91, 92, 93, 94, 95, 96, 81, 98, 97, 100
D6
6, 2, 3, 4, 5, 60, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 1, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 62, 200, 35, 38, 39, 36, 41, 37, 24, 40, 45, 42, 47, 44, 49, 50, 51, 52, 53, 54, 46, 56, 57, 58, 59, 43, 48, 55, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 61, 78, 79, 80, 81, 82, 83, 77, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 84, 97, 98, 99, 100
D7
7, 2, 3, 4, 5, 6, 70, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 1, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 72, 41, 42, 43, 44, 26, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 40, 60, 61, 62, 63, 64, 45, 66, 67, 68, 69, 65, 71, 59, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100
D8
8, 2, 3, 4, 5, 6, 7, 80, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 1, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 82, 200, 47, 45, 49, 50, 51, 52, 53, 54, 48, 56, 57, 58, 59, 60, 61, 62, 63, 64, 46, 66, 28, 68, 69, 70, 71, 72, 73, 74, 55, 76, 77, 78, 79, 67, 75, 81, 83, 84, 85, 65, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100
D9
9, 2, 3, 4, 5, 6, 7, 8, 90, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 1, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 92, 94, 100, 51, 130, 55, 52, 54, 56, 59, 57, 61, 58, 63, 60, 65, 62, 67, 64, 30, 70, 71, 72, 53, 74, 75, 76, 66, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 50, 69, 91, 89, 93, 73, 95, 96, 97, 98, 99, 68
____________________
November 19th update
Hans Havermann:
> I had a look at S9 because it seemed a challenge.
Here is my (hopefully correct) compute for the lexicographically earliest S9:
S9 = ninth, twentieth, thirtieth, fourth, ninetieth, sixth, seventh, eighth, first, tenth, second, twelfth, fifteenth, thirtyeighth, thirteenth, fifth, eleventh, eighteenth, nineteenth, seventeenth, twentyfirst, twentysecond, fourteenth, twentyfourth, twentyfifth, twentysixth, twentyseventh, twentyeighth, sixteenth, third, thirtyfirst, thirtysecond, thirtythird, thirtyfourth, thirtyfifth, thirtysixth, thirtyseventh, twentythird, thirtyninth, fortieth, fortyfirst, fortysecond, twentyninth, fortyfourth, fortyfifth, fortysixth, fortyseventh, fortyeighth, fortyninth, fortythird, fiftyfirst, fiftysecond, fiftythird, fiftieth, fiftyfifth, fiftyfourth, fiftyseventh, fiftyeighth, fiftyninth, sixtieth, sixtyfirst, sixtysecond, sixtythird, sixtyfourth, sixtyfifth, fiftysixth, sixtyseventh, sixtyeighth, sixtyninth, seventieth, seventyfirst, seventysecond, seventythird, seventyfourth, seventyfifth, seventysixth, seventyseventh, seventyeighth, sixtysixth, eightieth, eightyfirst, eightysecond, eightythird, eightyfourth, eightyfifth, eightysixth, eightyseventh, seventyninth, eightyninth, eightyeighth, ninetyfirst, ninetysecond, ninetythird, ninetyfourth, ninetyfifth, ninetysixth, ninetyseventh, ninetyeighth, ninetyninth, onehundredth, ...
i.e., the ordinals of 9, 20, 30, 4, 90, 6, 7, 8, 1, 10, 2, 12, 15, 38, 13, 5, 11, 18, 19, 17, 21, 22, 14, 24, 25, 26, 27, 28, 16, 3, 31, 32, 33, 34, 35, 36, 37, 23, 39, 40, 41, 42, 29, 44, 45, 46, 47, 48, 49, 43, 51, 52, 53, 50, 55, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 56, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 66, 80, 81, 82, 83, 84, 85, 86, 87, 79, 89, 88, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, ...
So that would be red entries for:
ninth
twentieth
thirtieth
ninetieth
first
second
fifteenth
thirtyeighth
thirteenth
fifth
eleventh
seventeenth
fourteenth
sixteenth
third
twentythird
twentyninth
fortythird
fiftieth
fiftyfourth
fiftysixth
sixtysixth
seventyninth
eightyeighth
... so 24 swaps.
____________________
Nov 24th update
Hans Havermann:
> I had a look at S10. Barring mistakes (and, as with look at S9, always using the smallest possible distinct integer ordinal), I calculated 11001 terms of it:
I make the 13th term to be "eleventhousandth", which results in a large number (1098) of integer ordinals that do not correspond to their cardinal positions (i.e. "swaps"). I've indicated these with an asterisk at the end of their lines. The final * is tenthousandninehundredeightysixth*
... meaning that from 11001 to infinity, every written-out ordinal is the ordinal of the index.
EA:
Hereunder is how the above (tenth).txt list by Hans starts (the successive yellow vertical terms are the successive horizontal terms of the list; the tenth letter of the list is a t, the eighteenth letter is a e, the twentyeighth is a n , the ninetieth letter is a t, the fifth letter is a h, etc.):
1 t tenth *
2 e eighteenth *
3 n twentyeighth *
4 t ninetieth *
5 h fifth
6 e second *
7 i seventh
8 g eighth
9 h ninth
10 t first *
11 e sixth *
12 e eleventh *
13 n eleventhousandth *
14 t fourth *
15 h fifteenth
16 t fourteenth *
17 w seventeenth
18 e twelfth *
19 n third *
20 t sixteenth *
21 y twentyfirst
22 e twentysecond
23 i twentythird
24 g twentyfourth
25 h twentyfifth
26 t twentieth *
27 h twentyseventh
28 n thirteenth *
29 i twentyninth
30 n nineteenth *
31 e thirtyfirst
32 t twentysixth *
33 i thirtythird
34 e thirtyfourth
35 t thirtysecond *
36 h thirtysixth
37 f thirtyseventh
38 i thirtyeighth
39 f thirtyninth
40 t thirtyfifth *
41 h fortyfirst
(...)
This page is mentioned by Greg Ross' Futility Closet here:
Many thanks to Giorgos and Hans!
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