Boucles d’Alexandre

(Alexandre’s loops/curls)

Happy birthday !

(March 28^th)

A=1, B=2, C=3, D=4, E=5, F=6, G=7, H=8, I=9, J=10, K=11, L=12, M=13, N=14, O=15, P=16, Q=17, R=18, S=19, T=20, U=21, V=22, W=23, X=24, Y=25, Z=26.

Start with a word or a name – here “ALEXANDRE”;

Write on top of each letter its rank in the alphabet:

1 12 5 24 1 14 4 18 5

A L E X A N D R E

Start with the first integer (here ‘1’), add or subtract the next integer (‘12’) in order to keep the result as close as possible to zero – but never under zero (we do here 1+12=13 as we cannot do 1-12=-11):

1 12 5 24 1 14 4 18 5

A L E X A N D R E

S = 1,13,

Proceed like this for the full word/name:

1+12-5+24 -1+14 -4+18 -5

A L E X A N D R E

S = 1,13,8,32,31,17,13,31,26

Duplicate the word/name and compute S accordingly – until it appears that S has entered a loop (in yellow here):

1+12-5+24 -1+14 -4+18 -5 -1-12-5+24 -1-14 -4+18 -5 -1

A L E X A N D R E A L E X A N D R E A...

S = 1,13,8,32,31,17,13,31,26,25,13,8,32,33,17,13,31,26,25, ...

The next example will raise a few questions:

5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3

E R I C E R I C E R I C E R I C E R I C E R I C E R I C E R I C

S = 5,23,14,11,6,24,15,12,7,25,16,13,8,26,17,14,9,27,18,15,10,28,19,16,11,29,20,17,12,30,21,18,

5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3

E R I C E R I C E R I C E R I C E R I C E R I C E R I C E R I C

S = 13,31,22,19,14,32,23,20,15,33,24,21,16,34,25,22,17,35,26,23,18, 0,9,6,1,19,10,7,2,20,11,8,

5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3 5 18 9 3

E R I C E R I C E R I C ...

S = 3,21,12,9,4,22,13,10,5,23,14,11,... (yellow loop length = 72)

Question 1:

Alexandre’s loop has 9 terms and Eric’s one 72: is there a quick way to anticipate these results, only by considering a few letters of the name/word?

Question 2:

What is the fate of the number names (in English)? Are there numbers N whose loop has N terms?

Question 3:

What are the numbers N which appear in their S sequence? ZERO is such a number:

26 5 18 15 26 5 18 15 26 5 18 15 26 5 18 15 26 5 18 15 26 5 18 15

Z E R O Z E R O Z E R O Z E R O Z E R O Z E R O

S = 26,21, 3,18,44,39,21, 6,32,27, 9,24,50,45,27,12,38,33,15, 0,26,21,... (20-term loop; 0 belongs to S)

Best,

É.

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Maximilian Hasler and Hans Havermann were quick to answer questions 2 and 3:

Maximilian, question 3, “zero” to “twenty” (YES = N belongs to its sequence S):

("zero") YES

S = 26,21,3,18,44,39,21,6,32,27,9,24,50,45,27,12,38,33,15,0,26,*** LOOP DETECTED

of length = 20

("one") YES

S = 15,1,6,21,7,2,17,3,8,23,9,4,19,5,0,15,*** LOOP DETECTED of length = 15

("two") NO

S = 20,43,28,8,31,16,36,13,28,*** LOOP DETECTED of length = 6

("three") NO

S = 20,12,30,25,20,0,8,26,21,16,36,28,10,5,0,20,*** LOOP DETECTED of length = 15

("four") NO

S = 6,21,0,18,12,27,6,24,18,3,24,6,0,15,36,18,*** LOOP DETECTED of length = 12

("five") NO

S = 6,15,37,32,26,17,39,34,28,19,41,36,30,21,43,38,32,23,1,6,0,9,31,26,20,11,33,

28,22,13,35,30,24,15,*** LOOP DETECTED of length = 32

("six") YES

S = 19,10,34,15,6,30,11,2,26,7,16,40,21,12,36,17,8,32,13,4,28,9,0,24,5,14,38,19,

*** LOOP DETECTED of length = 27

("seven") NO

S = 19,14,36,31,17,36,31,9,4,18,37,32,10,5,19,0,5,27,22,8,27,22,0,5,*** LOOP DETECTED

of length = 10

("eight") YES

S = 5,14,7,15,35,30,21,14,6,26,21,12,5,13,33,28,19,12,4,24,19,10,3,11,31,26,17,10,2,22,

17,8,1,9,29,24,15,8,0,20,15,6,13,5,25,20,11,4,12,32,27,18,11,3,23,18,9,2,10,30,25,

16,9,1,21,16,7,0,8,28,23,14,*** LOOP DETECTED of length = 70

("nine") YES

S = 14,5,19,14,0,9,23,18,4,13,27,22,8,17,3,8,22,13,*** LOOP DETECTED of length = 8

("ten") YES

S = 20,15,1,21,16,2,22,17,3,23,18,4,24,19,5,25,20,6,26,21,7,27,22,8,28,23,9,29,24,10,30,

25,11,31,26,12,32,27,13,33,28,14,34,29,15,35,30,16,36,31,17,37,32,18,38,33,19,39,34,

20,0,5,19,*** LOOP DETECTED of length = 6

("eleven") YES

S = 5,17,12,34,29,15,10,22,17,39,34,20,15,3,8,30,25,11,6,18,13,35,30,16,11,23,18,40,35,

21,16,4,9,31,26,12,7,19,14,36,31,17,12,0,5,27,22,8,3,15,10,32,27,13,8,20,15,37,32,18,

13,1,6,28,23,9,4,16,11,33,28,14,9,21,16,38,33,19,14,2,7,29,24,10,5,*** LOOP DETECTED

of length = 84

("twelve") NO

S = 20,43,38,26,4,9,29,6,1,13,35,30,10,33,28,16,38,33,13,36,31,19,41,36,16,39,34,22,0,5,

25,2,7,19,*** LOOP DETECTED of length = 12

("thirteen") NO

S = 20,12,3,21,1,6,1,15,35,27,18,0,20,15,10,24,4,12,*** LOOP DETECTED of length = 16

("fourteen") YES

S = 6,21,0,18,38,33,28,14,8,23,2,20,0,5,0,14,*** LOOP DETECTED of length = 8

("fifteen") YES

S = 6,15,9,29,24,19,5,11,2,8,28,23,18,4,10,1,7,27,22,17,3,9,0,6,26,21,16,2,8,17,11,31,26,

21,7,1,10,4,24,19,14,0,6,*** LOOP DETECTED of length = 42

("sixteen") NO

S = 19,10,34,14,9,4,18,37,28,4,24,19,14,0,19,*** LOOP DETECTED of length = 14

("seventeen") YES

S = 19,14,36,31,17,37,32,27,13,32,27,5,0,14,34,29,24,10,29,24,2,7,21,1,6,1,15,34,29,7,2,16,

36,31,26,12,31,26,4,9,23,3,8,3,17,36,31,9,4,18,38,33,28,14,33,28,6,1,15,35,30,25,11,30,

25,3,8,22,2,7,2,16,35,30,8,3,17,*** LOOP DETECTED of length = 72

("eighteen") NO

S = 5,14,7,15,35,30,25,11,6,15,8,0,20,15,10,24,19,10,3,11,31,26,21,7,2,11,4,12,32,27,22,8,3,

12,5,13,33,28,23,9,4,13,6,14,34,29,24,10,5,*** LOOP DETECTED of length = 48

("nineteen") YES

S = 14,5,19,14,34,29,24,10,24,15,1,6,26,21,16,2,16,7,21,16,36,31,26,12,26,17,3,8,28,23,18,4,18,

9,23,18,38,33,28,14,0,9,*** LOOP DETECTED of length = 8

("twenty") YES

S = 20,43,38,24,4,29,9,32,27,13,33,8,28,5,0,14,34,9,29,6,1,15,35,10,30,7,2,16,36,11,31,8,3,17,

37,12,32,9,4,18,38,13,33,10,5,19,39,14,34,11,6,20,0,25,5,28,23,9,29,4,24,1,6,*** LOOP

DETECTED of length = 12

Hans Havermann, question 3: What are the numbers N which appear in their S sequence?

The bracketed numbers are {length of S at end of the initial loop, position of N in S, length of the loop}

1 one {15,2,15} N is in the loop: 2, 17, 32, 47, ...

6 six {27,5,27} N is in the loop: 5, 32, 59, 86, ...

8 eight {75,32,16}

9 nine {20,6,5}

10 ten {63,30,6}

11 eleven {84,18,40}

14 fourteen {16,8,8}

15 fifteen {42,2,19}

17 seventeen {81,5,3}

19 nineteen {48,3,8}

20 twenty {66,1,12}

21 twentyone {36,13,18}

22 twentytwo {63,22,54} N is in the loop: 22, 76, 130, 184, ...

23 twentythree {77,46,22}

24 twentyfour {30,4,20}

25 twentyfive {90,53,5}

26 twentysix {72,36,72} N is in the loop: 36, 108, 180, 252, ...

27 twentyseven {55,9,38}

28 twentyeight {22,20,2}

29 twentynine {70,6,36}

32 thirtytwo {36,24,8}

34 thirtyfour {50,16,12}

38 thirtyeight {165,138,2}

46 fortysix {80,69,72} N is in the loop: 69, 141, 213, 285, ...

48 fortyeight {80,5,14}

The position of 22 in S of twentytwo is 22.

Hans Havermann, question 2: « What is the fate of the number names (in English)? Are there numbers N whose loop has N terms? »

I used a very old Mathematica routine that creates English number words from numbers and altered it to remove all spaces and hyphens:

NumberName[12345678901234567890123456789]

Twelveoctillionthreehundredfortyfiveseptillionsixhundredseventyeightsextillionninehundred

Onequintilliontwohundredthirtyfourquadrillionfivehundredsixtyseventrillioneighthundrednin

etybilliononehundredtwentythreemillionfourhundredfiftysixthousandsevenhundredeightynine

I then created a program to test these words for your property.

I have three solutions (number, NumberName, loop):

12 twelve {16,39,34,22,0,5,25,2,7,19,41,36}

30 thirty {20,12,3,21,1,26,6,14,5,23,3,28,8,0,9,27,7,32,12,4,13,31,11,36,16,8,17,35,15,40}

56 fiftysix {6,15,9,29,4,23,14,38,32,23,17,37,12,31,22,46,40,31,25,5,30,11,2,26,20,11,5,25,0,19,10,

34,28,19,13,33,8,27,18,42,36,27,21,1,26,7,16,40,34,25,19,39,14,33,24,0}

It’s unlikely that there are more, but I’ll let it run a bit.

_____________________________

Thank you Maximilian and Hans!

Best,

É.