Surface of a number

 

To define the surface S of a number N we form two substrings of digits b and h with a single cut through N; we then compute the product b*h (base*height = S).
 
The surface of 8 is zero (as b or h are missing).
The surface of 10 is also zero (as b = 1 and h = 0, for instance).
The surface of 11 is 1 (as 1*1 = 1).
The surface of 39 is 27 (as 3*9 = 27).
2021 has two distinct surfaces, depending on where you cut:  the cut 202/1 will produce S = 202*1 = 202 and the cut 20/21 will produce S = 20*21 = 420 (no substring can start with a leading zero, except the substring 0 itself).
1234567890 has 9 distinct surfaces (9 possible cuts), etc.
 
The interesting part is that every number is the surface of at least two other numbers;
17 (a prime number) is one of the surfaces of 117 (1*17) and one of the surfaces of 171 (17*1);
2021 is one of the surfaces of 4347 (as 43*47 = 2021) but also one of the surfaces of 12021 and 20211 (as 2021 = 1*2021 = 2021*1).
 
Almost all numbers > 111 have more than one surface. Those surfaces can be ordered from the smallest to the largest; 112 has two surfaces for instance: 1*12 = 12 and 11*2 = 22. We call the smallest (12) the SurfMin and the largest (22) the SurfMax.
 
2021 has SurfMin = 202 (= 202*1) and SurfMax = 420 (= 20*21);
1234567890 has SurfMin = 0 (= 123456789*0) and SurfMax = 1111111020 (= 12345678*90) .
 
A difficult task is to find an integer that produces a given surface.
For instance, what is the smallest integer with SurfMin = 108? Here is our modus operandi:
 
First we decompose 108 into 2*2*3*3*3;
as we have only two factors (b and h) to find, we list all possible pairs such that b*h = 108:
2*54
3*36
4*27
6*18
9*12
But also, swapping b and h:
12*9
18*6
27*4
36*3
54*2

We now arrange from smallest to biggest the above integers:
129, 186, 254, 274, 336, 363, 427, 542, 618, 912.
Properly cut, those integers will all lead to 108.
But as we want 108 to be the smallest possible product b*h (or h*b), we must eliminate all the products that are < 108, as they would ruin the solution we are looking for:

129 has a SurfMin = 1*29 = 29, which doesn’t fit (as 29 < 108);
186 has a SurfMin of 86 (= 1*86), which doesn’t fit either (86 < 108);
254 has a SurfMin of 100 (= 25*4), which still doesn’t fit (100 < 108);
274 has a SurfMin of 108 (= 27*4), which is ok, we have a hit.
The answer to the question: “What is the smallest integer with SurfMin = 108” is 274.
 
Let’s try to find a few sequences (my apologizes for the inevitable typos hereunder):
(Many thanks to Hans Havermann and Maximilian Hasler for their corrections; Maximilian has noticed that the sequence B is already in the OEIS, here)
In our next post we will talk about the volume of a number  !-)


Commentaires

Enregistrer un commentaire

Posts les plus consultés de ce blog

Beautés ?

Image
Beau comme un clavier ? Un deuxième ? (celui de Modotti)   Beau comme le dos d ’ un timbre proposé par Delcampe ? Beau comme une série ukrainienne où les avions râpent le Kremlin ? Belle comme Chargois ? ... belle comme Germaine Richier ? Bel atelier ? Beau comme une pièce de 1 santim ? ... ou comme une pièce de 50  santeem ? ... Byrrh , beau comme un anus ? Beau comme la collection de Marin Karmitz à Beaubourg ? Belles Ménines ? Beau comme Marcel Duchamp ? Beau comme Jean-Luc Godard ? Beau comme la réouverture de l ’ Einsteinturm à Berlin ? Beau comme un anneau dudit ? Beau comme Eisenstein ? Beau comme les fleurs de Pascale hier matin ? Beau comme on dirait la planète Mars ? Beau comme des recherches personnelles ? Beau comme une marelle (vers le ciel) ? Beau comme un triple selfie (au Man Ray ) ? Beau comme le ciel de Sacha , mardi, en face du restaurant La Montagne ? (qui a rouvert) ... beau comme ce bleu-là de Lo ? Beau comme le froid ? Beau comme une Vespa ? Beau comme Cloon
Lire la suite

Underline, reproduce

Image
Dall.e creation If we underline the... « first, second, third, fourth, fifth, sixth, seventh, eighth, ninth, tenth, eleventh,… » letter of this sentence S 1 , we see that the successive underlined letters simply reproduce S 1 . No big deal. If we decide not to start S 1   with "first" but with "sixth", we can reproduce the new S 6  quite easily: S 6 = sixth , second, third, fourth, fifth, first , seventh, eighth, ninth, tenth, eleventh,… S 6  was reproduced in a slightly less mechanical manner than  S 1 :  just two words of  S 1   swapped here ("first" and "sixth", in red, above). If we start S 8  with "eighth" we just swap again a pair of words of S 1  ("first" and "eighth", in red, hereunder): S 8  = eighth , second, third, fourth, fifth, sixth, seventh, first , ninth, tenth, eleventh,… What about starting S 9  with "ninth"? How many swaps do we need to reproduce S 9 ?  This is more tricky! (Indeed, we
Lire la suite

Le tripalin se présente

Image
Un tripalin est constitué de trois images. Chaque image illustre un substantif. Accolés, ces trois substantifs forment une chaîne palindromique. Laquelle nous vous invitons à trouver.  Exemple ci-dessous. Le premier mot est COL-BLEU Le second DUEL Le troisième BLOC . Ces mots forment la chaîne COLBLEUDUELBLOC qui est bien palindromique. Je posterai chaque jeudi une nouvelle salve de tripalins . Rien que des noms illustrés – pas d’adjectifs, pas de verbes conjugués, pas d’adverbes (parfois un prénom ou un nom propre – mais seulement quand il faut). À la demande générale (de mon neveu), chaque tripalin sera surmonté de son numéro d ’ ordre ( octothorpe noir) et de trois nombres bleus. Ces nombres reprennent les trois longueurs (en lettres) des trois mots présents dans le tripalin. On a par exemple ci-dessous [ #0 ↓ 7+4+4 ] car COL-BLEU comporte  7  lettres, DUEL   4  et BLOC   4 . Le numéro d ’ ordre est zéro. #0 ↓ 7+4+4 Voilà – tous retours critiques, construits ou déconstruits, bien
Lire la suite