The box ■ operation


Hello SeqFans,
Let's define  (a ■ b) = c  with an example:

     12951
 ■   2019
 --------
 =  10948 

 We align a and b on the right and make the absolute differences of the vertically disposed digits. For example, the 8 above comes from 1 - 9 and the 4 from 5 - 1.
The result 10948 starts with 1 as this 1 comes from 1 - 0 (the 0 being "invisible" though).

 The box algebra is fun to explore – note for instance that one always has (a ■ b) = (b ■ a), and (a ■ b) ■ c = a ■ (b ■ c), etc. [NO for the associativity! Read Maximilian below!]

 Here is a sequence S of numbers such as (n ■ k) is always a square, k being the smallest possible integer:

 S = 2, 1, 2, 3, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 5, 6, 7, 8, 9, 21, 1, 2, 3, 4, 6, 7, 8, 9, 19, 10, 11, 1, 2, 3, 9, 17, 18, 19, 29, 20, 10, 11, 12, 13, 19, 27, 28, 29, 39, 30, 20, 21, 22, 10, 4, 5, 6, 7, 8, 1, ...

 Example:
For n = 1 the smallest k producing a square is 2 (as 1 ■ 2 = 1, this 1 being the square of 1);
For n = 2 the smallest k producing a square is 1 (as 2 ■ 1 = 1, this 1 being the square of 1);
For n = 3 the smallest k producing a square is 2 (as 3 ■ 2 = 1, this 1 being the square of 1);
For n = 4 the smallest k producing a square is 3 (as 4 ■ 3 = 1, this 1 being the square of 1);
For n = 5 the smallest k producing a square is 3 (as 5 ■ 1 = 4, this 4 being the square of 2);
For n = 6 the smallest k producing a square is 3 (as 6 ■ 2 = 4, this 4 being the square of 2);
...
For n = 16 the smallest k producing a square is 12 (as 16 ■ 12 = 4, this 4 being the square of 2);
For n = 17 the smallest k producing a square is 1 (as 17 ■ 1 = 16, this 16 being the square of 4);
etc.

 My friend Jean-Marc Falcoz has computed the first 20000 terms of S. He writes me that the highest k is 2175 so far (with 16575 ■ 2175 = 14400, square of 120) – the missing ks so far being 565, 678, 680, 681, etc.

 The graph of S is emblematic of the rainfalls that affect Brussels for at least another week!


Have fun until the ■ Boxing Day!
Best,
É.


Commentaires

  1. MFH6 décembre 2019 à 05:38

    Very funny! Yes, the ■ operation has the nice properties of being commutative, associative, and having a neutral element e = 0 (x ■ 0 = 0 ■ x = x) and every element x has a symmetric x' such that x ■ x' = e, namely itself: x' = x. This makes it an abelian group. So it is natural to think of it as a fancy law of addition. One might want to find a "compatible" multiplication, i.e., distributive over ■. Then it might be natural to consider squares for that multiplication, rather than the usual one.
    (The property x+x=0 holds in [https://en.wikipedia.org/wiki/Boolean_ring Boolean rings] characterized by x² = x for all x. But in that case, "squares" are obviously not very interesting...)

    RépondreSupprimer
    Réponses
    1. MFH6 décembre 2019 à 06:25

      Oops, actually ■ is *not* associative as you wrote [(a ■ b) ■ c =?= a ■ (b ■ c)], consider e.g., a=2, b=c=1. [Else you could find the a(n)=k you are looking for as min { x² ■ n, y² ■ n} \ {0}, where x² is the largest square < n and y² is the smallest square > n, I think.]

      Supprimer
    2. Éric ANGELINI6 décembre 2019 à 08:06

      Geeee, glad there is someone reading my posts! Thank you Maximilian!

      Supprimer

Enregistrer un commentaire

Posts les plus consultés de ce blog

Confingame, 3e étape

Image
Sur le dernier plateau-repas de Timothy McVeigh il y avait un litre de crème glacée à la menthe et aux pépites de chocolat. Sur celui de Victor Feguer, une olive unique non dénoyautée – il pensait qu’elle allait germer dans son corps et donner un olivier, « symbole de paix ». John Wayne Gacy fit honneur à son métier (gérant de trois KFC) en commandant des crevettes grillées, du poulet grillé, des frites et des fraises. Ses derniers mots furent Kiss my ass . Sacco et Vanzetti eurent droit à de la soupe, de la viande, des toasts et du thé. Le plateau-repas d’Angel Nieves Diaz resta vide, car il ne voulait rien. On lui apporta le plat réglementaire servi en prison ce jour-là (dinde hachée avec assaisonnement pour tacos, riz, fromage râpé, haricots mouchetés, tortillas, croustillant aux pommes, Ice Tea ), mais il ne toucha à rien*. _____________ * Murderpedia .  Contributeurs à la 3e étape du Confingame : Pascale Angelini-Cordonnier propose une captur...
Lire la suite

Square my chunks and add

Image
The idea is to transform the integer a into b and iterate. Cut a into the number of chunks you want, square them, add those squares. Example a = 1023 can be transformed in b  (last column) like this: 1 ² + 0 ² + 2 ² + 3 ² = 1 + 0 + 4 + 9  = 14 = b 1 ² + 0 ² + 23 ²     = 1 + 0 + 529    = 530   10 ² + 2 ² + 3 ²     = 100 + 4 + 9    = 113   10 ² + 23 ²         = 100 + 529      = 629   102 ² + 3 ²         = 10404 + 9      = 10413   1023 ²                              = 1046529   Question #1 Starting with 2 and iterating, what is the shortest path ending in 1? Here are 11 steps (left column) – can you do better? 2    –> 2² = 4 4    –> 4² = 16 16   –> 1² + 6² = 1 + 36 = 37 37   –> 37² = 1369...
Lire la suite

A square for three (chess)

Image
(English translation after the French text) Voici cinq problèmes d’échecs disjoints : a ) combien faut-il de coups au minimum pour que trois pions soient capturés sur la même case ? b ) trois tours c ) trois cavaliers d ) trois fous e ) trois dames Here are five disjointed chess problems: a) how many moves are required at least for three pawns to be captured on the same square? b) three rooks c) three knights d) three bishops e) three queens Solutions a ) (corrigé grâce à Orion , ici ) 3,5 coups pour 3 pions capturés en b5 (4 moves for White and 3 moves for Black   — 3 pawns  captured on b5)  b ) 5  coups pour 3 tours capturées en a3 (ci-dessous – merci aussi à Orion !-) (5 moves for White and 5 moves for Black  — 3 rooks  captured on a3  hereunder )  c ) 5  coups pour 3 cavaliers capturés en d5  ( ci-dessous) (5 moves for White and 5 moves for Black   — 3 knights  captured on d5  hereunder )  (merci Orion !-) d ) ...
Lire la suite