Procust and the digits

(English translation provided by Google)

(original French text at the end of the post)

We can no longer tolerate inequality. Numbers. And we decided to shave everything that goes beyond in order to restore a semblance of order.

Let N be an integer with k digits.

We add up the digits of N.

We divide the sum by k.

We replace each digit of N with this result.

If necessary, we need to round some results to the nearest integer – upwards or downwards.

We want the sum of these results, rounded or not, to be the same as the original one.

That's all.

We have thus transformed N into a new number M, which includes always k digits — and these k digits have the same sum as before.

M must always start with its smallest digits (but not with zero).

None of the digits of M now exceed another by more than one unit.

The world is less unequal.

What is the value of the sequence of successive M for N = 1, 2, 3, 4, 5, …?

Here is what N = 2024 becomes for example:

1) 2+0+2+4 = 8

2) 8/4 = 2

3) M = 2222

Here is what N = 2025 becomes:

1) 2+0+2+5 = 9

2) 9/4 = 2.25

3) M = 2223

The numbers 0 to 12 remain themselves – then:

13 —> 22

14 —> 23

15 —> 33

16 —> 34

17 —> 44

18 —> 45

19 —> 55

20 —> 11

21 —> 12

22 —> 22

23 —> 23

25 —> 34

26 —> 44

…

Is the graph of the first 10,000 M fractal?

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Jean-Marc Falcoz was quick to compute M.

> I have attached a list of the first 1000 terms. And... yes, the graph is indeed fractal (see below images for 100, 1000, and 10000 terms)

M = {1,2,3,4,5,6,7,8,9,10,11,12,22,23,33,34,44,45,55,11,12,22,23,33,34,44,45,55,56,12,22,23,33,34,44,45,55,56,66,22,23,33,34,44,45,55,56,66,67,23,33,34,44,45,55,56,66,67,77,33,34,44,45,55,56,66,67,77,78,34,44,45,55,56,66,67,77,78,88,44,45,55,56,66,67,77,78,88,89,45,55,56,66,67,77,78,88,89,99,100,101,111,112,122,222,223,233,333,334,101,111,112,122,222,223,233,333,334,344,111,112,122,222,223,233,333,334,344,444,112,122,222,223,233,333,334,344,444,445,122,222,223,233,333,334,344,444,445,455,222,223,233,333,334,344,444,445,455,555,223,233,333,334,344,444,445,455,555,556,233,333,334,344,444,445,455,555,556,566,333,334,344,444,445,455,555,556,566,666,334,344,444,445,455,555,556,566,666,667,101,111,112,122,222,223,233,333,334,344,111,112,122,222,223,233,333,334,344,444,112,122,222,223,233,333,334,344,444,445,122,222,223,233,333,334,344,444,445,455,222,223,233,333,334,344,444,445,455,555,223,233,333,334,344,444,445,455,555,556,233,333,334,344,444,445,455,555,556,566,333,334,344,444,445,455,555,556,566,666,334,344,444,445,455,555,556,566,666,667,344,444,445,455,555,556,566,666,667,677,111,112,122,222,223,233,333,334,344,444,112,122,222,223,233,333,334,344,444,445,122,222,223,233,333,334,344,444,445,455,222,223,233,333,334,344,444,445,455,555,223,233,333,334,344,444,445,455,555,556,233,333,334,344,444,445,455,555,556,566,333,334,344,444,445,455,555,556,566,666,334,344,444,445,455,555,556,566,666,667,344,444,445,455,555,556,566,666,667,677,444,445,455,555,556,566,666,667,677,777,112,122,222,223,233,333,334,344,444,445,122,222,223,233,333,334,344,444,445,455,222,223,233,333,334,344,444,445,455,555,223,233,333,334,344,444,445,455,555,556,233,333,334,344,444,445,455,555,556,566,333,334,344,444,445,455,555,556,566,666,334,344,444,445,455,555,556,566,666,667,344,444,445,455,555,556,566,666,667,677,444,445,455,555,556,566,666,667,677,777,445,455,555,556,566,666,667,677,777,778,122,222,223,233,333,334,344,444,445,455,222,223,233,333,334,344,444,445,455,555,223,233,333,334,344,444,445,455,555,556,233,333,334,344,444,445,455,555,556,566,333,334,344,444,445,455,555,556,566,666,334,344,444,445,455,555,556,566,666,667,344,444,445,455,555,556,566,666,667,677,444,445,455,555,556,566,666,667,677,777,445,455,555,556,566,666,667,677,777,778,455,555,556,566,666,667,677,777,778,788,222,223,233,333,334,344,444,445,455,555,223,233,333,334,344,444,445,455,555,556,233,333,334,344,444,445,455,555,556,566,333,334,344,444,445,455,555,556,566,666,334,344,444,445,455,555,556,566,666,667,344,444,445,455,555,556,566,666,667,677,444,445,455,555,556,566,666,667,677,777,445,455,555,556,566,666,667,677,777,778,455,555,556,566,666,667,677,777,778,788,555,556,566,666,667,677,777,778,788,888,223,233,333,334,344,444,445,455,555,556,233,333,334,344,444,445,455,555,556,566,333,334,344,444,445,455,555,556,566,666,334,344,444,445,455,555,556,566,666,667,344,444,445,455,555,556,566,666,667,677,444,445,455,555,556,566,666,667,677,777,445,455,555,556,566,666,667,677,777,778,455,555,556,566,666,667,677,777,778,788,555,556,566,666,667,677,777,778,788,888,556,566,666,667,677,777,778,788,888,889,233,333,334,344,444,445,455,555,556,566,333,334,344,444,445,455,555,556,566,666,334,344,444,445,455,555,556,566,666,667,344,444,445,455,555,556,566,666,667,677,444,445,455,555,556,566,666,667,677,777,445,455,555,556,566,666,667,677,777,778,455,555,556,566,666,667,677,777,778,788,555,556,566,666,667,677,777,778,788,888,556,566,666,667,677,777,778,788,888,889,566,666,667,677,777,778,788,888,889,899,333,334,344,444,445,455,555,556,566,666,334,344,444,445,455,555,556,566,666,667,344,444,445,455,555,556,566,666,667,677,444,445,455,555,556,566,666,667,677,777,445,455,555,556,566,666,667,677,777,778,455,555,556,566,666,667,677,777,778,788,555,556,566,666,667,677,777,778,788,888,556,566,666,667,677,777,778,788,888,889,566,666,667,677,777,778,788,888,889,899,666,667,677,777,778,788,888,889,899,999,1000}

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French version (and the story of Procust)

Nous ne supportons plus les inégalités. De chiffres. Et avons décidé de raser tout ce qui dépasse afin de remettre un semblant d’ordre.

Soit N un entier ayant k chiffres.

On fait la somme des chiffres de N.

On la divise par k.

On remplace chaque chiffre de N par ce résultat.

Si nécessaire, nous devons arrondir certains résultats à l’entier le plus proche – vers le haut ou vers le bas.

Nous voulons en effet que la somme de ces résultats, arrondis ou non, soit la même que celle d’origine.

C’est tout.

Nous avons ainsi transformé N en un nouveau nombre M, lequel comporte toujours k chiffres — et ces k chiffres ont la même la somme qu’avant.

M doit toujours commencer par ses plus petits chiffres (mais pas par zéro).

Aucun des chiffres de M n’en dépasse désormais un autre par plus d’une unité.

Le monde est moins inégal.

Que vaut la suite des M successifs pour N = 1, 2, 3, 4, 5, … ?

Voici ce que devient par exemple N = 2024 :

1) 2+0+2+4 = 8

2) 8/4 = 2

3) M = 2222

Voici ce que devient par exemple N = 2025 :

1) 2+0+2+5 = 9

2) 9/4 = 2.25

3) M = 2223

Les nombres de 0 à 12 restent eux-mêmes, puis :

13 —> 22

14 —> 23

15 —> 33

16 —> 34

17 —> 44

18 —> 45

19 —> 55

20 —> 11

21 —> 12

22 —> 22

23 —> 23

25 —> 34

26 —> 44

…

Le graphique des 10 000 premiers M est-il fractal ?

____________________

Analyse procustréenne

Le problème de Procuste

Procuste, personnage de la mythologie grecque, était un brigand qui forçait ses victimes à s’allonger sur son lit et leur étirait les jambes quand leurs pieds ne touchaient pas le bout du lit, ou au contraire leur coupait les jambes à la hache si elles dépassaient du lit.

La version scientifique de l’histoire, dans le domaine de l’analyse des formes, consiste à se donner pour lit une forme de référence, dont les propriétés sont connues, sur lequel on allongera des victimes venant d’un ensemble d’objets à étudier. Le problème consiste à comparer la forme du lit à la forme des victimes, opération difficile tant que l’une ne peut pas s’allonger sur l'autre.

Pour retailler les victimes et résoudre le problème, les scientifiques vont utiliser, en fait de hache, la translation, qui permettra de déplacer les victimes jusqu’au centre du lit, l’homothétie, qui changera la taille de la victime pour qu’elle soit égale à la taille du lit et enfin la rotation, qui trouvera la position la plus confortable pour la victime.

Il apparaît ainsi que la hache ne peut changer la forme de l’objet (toutes les transformations utilisées conservent les angles). Cela permet par exemple en biologie de comparer la forme du crâne d’un dauphin et d’un rat, en éliminant la différence de taille entre ces deux crânes, ainsi que les différences introduites lors de la numérisation des données (position et orientation sur l’appareil de mesure, par exemple).