Esto se basa en otro hilo . Para $m,n\in \mathbb N$ , dejemos que $f(m,n)$ sea el número mínimo de cuadrados con lados enteros necesarios para embaldosar un $m\times n$ rectángulo. Recientemente, una tabla de valores para $n\le m\le 85$ obtenido por lo que parece ser una búsqueda de fuerza bruta, ha sido puesto en línea ici .
La tabla parece bastante difusa, pero si la restringimos a valores de coprima $m,n$ tal que $2n\ge m\ge n$ La fluctuación es sorprendentemente escasa. Para mayor comodidad, la siguiente tabla ofrece $f(m,n)$ en orden inverso, mostrando en la fila $n$ los valores de $m=n-1,n-2,...,\lbrace n/2\rbrace$ pero poniendo "o" donde sea $(m,n)>1$ .
(El número que sigue a $m$ es $ g(m):=\frac{\log(m\sqrt{5})}{\log(\phi)}$ donde $\phi=\frac{\sqrt{5}+1}2$ ).
3 : 3.955 [ 3]
4 : 4.553 [ 4]
5 : 5.016 [ 5, 4]
6 : 5.395 [ 5, o]
7 : 5.716 [ 5, 5, 5]
8 : 5.993 [ 7, o, 5]
9 : 6.238 [ 7, 6, o, 6]
10 : 6.457 [ 6, o, 6, o]
11 : 6.655 [ 6, 7, 6, 6, 6]
12 : 6.836 [ 7, o, o, o, 6]
13 : 7.002 [ 7, 6, 7, 7, 6, 6]
14 : 7.156 [ 7, o, 7, o, 7, o]
15 : 7.299 [ 7, 8, o, 7, o, o, 8]
16 : 7.433 [ 7, o, 8, o, 7, o, 7]
17 : 7.559 [ 8, 8, 7, 8, 7, 7, 7, 8]
18 : 7.678 [ 8, o, o, o, 7, o, 7, o]
19 : 7.791 [ 7, 9, 7, 7, 7, 7, 7, 7, 7]
20 : 7.897 [ 9, o, 7, o, o, o, 7, o, 8]
21 : 7.999 [ 8, 7, o, 9, 8, o, o, 7, o, 7]
22 : 8.095 [ 8, o, 8, o, 8, o, 8, o, 8, o]
23 : 8.188 [ 8, 8, 8, 9, 8, 8, 8, 8, 8, 8, 8]
24 : 8.276 [ 8, o, o, o, 9, o, 8, o, o, o, 7]
25 : 8.361 [ 8, 8, 8, 8, o, 8, 8, 9, 8, o, 8, 8]
26 : 8.442 [ 8, o, 8, o, 8, o, 8, o, 9, o, 8, o]
27 : 8.521 [ 8,10, o, 8, 8, o, 8, 8, o, 8, 8, o, 8]
28 : 8.596 [ 8, o,10, o, 9, o, o, o, 8, o, 8, o, 8]
29 : 8.669 [ 9, 8, 8,10,10, 9, 9, 8, 9, 8, 8, 8, 9, 8]
30 : 8.740 [ 9, o, o, o, o, o, 9, o, o, o, 8, o, 9, o]
31 : 8.808 [ 8, 8, 8,10, 8, 8, 8, 8, 8, 8, 9, 8, 8, 8, 8]
32 : 8.874 [ 9, o, 8, o, 8, o, 9, o, 9, o, 8, o, 8, o, 9]
33 : 8.938 [ 9, 9, o, 9, 8, o, 8,10, o, 9, o, o, 8, 8, o, 9]
34 : 9.000 [ 9, o, 9, o, 9, o, 9, o, 8, o, 9, o, 8, o, 8, o]
35 : 9.060 [ 8, 9,10, 8, o, 9, o, 9, 8, o, 8, 9, 9, o, o, 8, 9]
36 : 9.119 [ 9, o, o, o,10, o,10, o, o, o, 9, o, 9, o, o, o,10]
37 : 9.176 [ 9, 9, 9, 9, 8,10, 9, 8, 9, 9, 9, 9, 8, 9, 8, 9, 8, 8]
38 : 9.231 [ 9, o, 9, o, 9, o, 9, o,10, o, 9, o, 9, o, 9, o,10, o]
39 : 9.285 [ 9, 9, o, 9, 9, o,10, 9, o, 9, 9, o, o, 9, o, 9, 9, o, 9]
40 : 9.338 [ 9, o, 9, o, o, o, 9, o, 9, o, 8, o, 9, o, o, o, 9, o, 8]
41 : 9.389 [ 9, 9, 9, 9,11, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9]
42 : 9.439 [ 9, o, o, o, 9, o, o, o, o, o,10, o,10, o, o, o,10, o,10, o]
43 : 9.488 [ 9, 9, 9, 9, 9, 9,10,10, 9, 9, 9, 9, 9, 9, 9, 9,10, 9, 9, 9, 9]
44 : 9.536 [ 9, o, 9, o, 9, o, 9, o, 9, o, o, o, 9, o, 9, o, 9, o, 9, o, 9]
45 : 9.582 [10, 9, o,10, o, o, 9, 9, o, o, 9, o, 9, 9, o,10, 9, o, 9, o, o, 9]
46 : 9.628 [ 9, o, 9, o, 9, o, 9, o, 9, o, 9, o, 9, o, 9, o, 9, o, 9, o, 9, o]
47 : 9.673 [ 9, 9, 9,11, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9,10, 9, 9, 9, 9, 9, 9, 9, 9]
48 : 9.716 [10, o, o, o, 9, o, 9, o, o, o,10, o,10, o, o, o, 9, o, 9, o, o, o, 9]
49 : 9.759 [ 9, 9, 9, 9,10,10, o, 9, 9, 9, 9, 9, 9, o, 9,10,10, 9, 9,10, o, 9, 9, 9]
50 : 9.801 [10, o, 9, o, o, o, 9, o, 9, o, 9, o, 9, o, o, o, 9, o, 9, o, 9, o, 9, o]
51 : 9.842 [ 9,10, o, 9,10, o,10, 9, o,10,10, o, 9,10, o, 9, o, o, 9,10, o,10,10, o, 9]
52 : 9.883 [10, o,11, o, 9, o,10, o,10, o, 9, o, o, o, 9, o,10, o, 9, o, 9, o,10, o,11]
53 : 9.922 [10, 9, 9,11, 9,10,10, 9,10,11,10, 9,10, 9,10,10,11,10, 9, 9, 9, 9, 9,11,11, 9]
54 : 9.961 [10, o, o, o, 9, o,10, o, o, o, 9, o, 9, o, o, o, 9, o, 9, o, o, o, 9, o,10, o]
55 : 9.999 [10,10,10,10, o, 9, 9, 9, 9, o, o, 9,10, 9, o, 9, 9, 9,10, o, 9, o,10, 9, o, 9, 9]
56 : 10.03 [ 9, o, 9, o,10, o, o, o, 9, o,10, o,11, o,10, o,10, o, 9, o, o, o,10, o, 9, o, 9]
57 : 10.07 [10,10, o,10, 9, o,10,10, o,10, 9, o, 9,10, o,10,10, o, o, 9, o,10, 9, o,10, 9, o,10]
58 : 10.11 [10, o,10, o,10, o,10, o,10, o,11, o,10, o,10, o,10, o,11, o,10, o,10, o,10, o,10, o]
59 : 10.14 [10,10, 9, 9,10,11, 9, 9, 9,10,10, 9, 9,10, 9, 9, 9,10, 9, 9,10, 9,10, 9, 9, 9, 9, 9,10]
60 : 10.18 [10, o, o, o, o, o,11, o, o, o,10, o,10, o, o, o,11, o, 9, o, o, o,10, o, o, o, o, o, 9]
61 : 10.21 [10, 9,10,10,10,10, 9, 9,10,10, 9,11,10,10,10,10, 9, 9,10,10, 9,10, 9, 9,10, 9,10, 9, 9, 9]
62 : 10.24 [10, o,10, o,10, o,10, o,10, o,10, o,11, o,11, o,10, o,11, o,10, o,10, o,10, o,10, o,10, o]
63 : 10.28 [10,10, o,10,10, o, o,10, o,10,10, o,10, o, o,10,10, o,10,10, o,10,10, o,10,10, o, o,10, o,10]
64 : 10.31 [10, o,10, o,10, o,10, o,11, o,11, o,10, o,10, o,10, o,10, o,10, o,10, o,10, o,10, o,10, o,10]
65 : 10.34 [10,10,10,10, o,10,10,10,10, o,10,10, o,10, o,11,10,10,10, o,10,10,10,10, o, o,10,10,11, o,10,10]
66 : 10.37 [10, o, o, o,10, o,10, o, o, o, o, o,10, o, o, o, 9, o,10, o, o, o,10, o,10, o, o, o, 9, o, 9, o]
67 : 10.40 [10,10,10,10,10,10,11,10,10,10,10,11,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,10,10,10,10]
68 : 10.44 [10, o,10, o,10, o,10, o,10, o,10, o,10, o,10, o, o, o,10, o,10, o,10, o,10, o,10, o,10, o,11, o,10]
69 : 10.47 [10,10, o,10,11, o,10, 9, o,10,10, o,10,11, o,11,11, o,10,11, o,10, o, o,10,10, o, 9, 9, o,10, 9, o, 9]
70 : 10.50 [11, o,10, o, o, o, o, o,10, o,10, o,10, o, o, o,11, o,10, o, o, o,10, o, o, o,10, o,10, o,10, o,10, o]
71 : 10.53 [10,10,10,12,10,10,10,10,10,11,10,12,10,10,10,11,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,10,10,10]
72 : 10.55 [10, o, o, o,10, o,10, o, o, o,10, o,10, o, o, o,10, o,10, o, o, o,10, o,10, o, o, o,10, o,10, o, o, o,10]
73 : 10.58 [10,10,10,11,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10]
74 : 10.61 [10, o,10, o,10, o,10, o,10, o,10, o,11, o,11, o,10, o,11, o,10, o,10, o,10, o,10, o,11, o,10, o,10, o,10, o]
75 : 10.64 [10,11, o,10, o, o,10,10, o, o,10, o,10,10, o,10,10, o,10, o, o,10,11, o, o,10, o,10,10, o,10,10, o,10, o, o,10]
76 : 10.67 [10, o,10, o,10, o,11, o,10, o,10, o,10, o,10, o,10, o, o, o,10, o,10, o,10, o,10, o,10, o,10, o,10, o,10, o,10]
77 : 10.69 [10,10,10,10,10,11, o,10,10,12, o,10,11, o,10,11,12,10,10,10, o, o,10,12,12,10,10, o,10,10,10,10, o,10, o,12,10,10]
78 : 10.72 [11, o, o, o,10, o,10, o, o, o,10, o, o, o, o, o,11, o,10, o, o, o,11, o,11, o, o, o,11, o,10, o, o, o,11, o,10, o]
79 : 10.75 [11,10,10,11,11,10,10,10,10,11,11,10,10,10,10,10,10,11,10,10,10,10,10,10,10,10,11,11,10,10,10,11,10,10,10,11,10,10,10]
80 : 10.77 [10, o,10, o, o, o,10, o,10, o,12, o,10, o, o, o,10, o,11, o,11, o,10, o, o, o,10, o,11, o,10, o,10, o, o, o,10, o,10]
81 : 10.80 [10,10, o,12,11, o,10,10, o,10,10, o,10,12, o,10,10, o,10,10, o,10,10, o,10,10, o,11,11, o,10,11, o,10,10, o,10,10, o,10]
82 : 10.82 [10, o,11, o,11, o,11, o,10, o,11, o,11, o,10, o,10, o,11, o,10, o,10, o,10, o,10, o,11, o,10, o,11, o,10, o,10, o,10, o]
83 : 10.85 [10,10,10,10,10,11,10,10,10,10,10,10,10,10,11,10,10,11,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,10,10,10,10,10,10]
84 : 10.87 [10, o, o, o,10, o, o, o, o, o,10, o,10, o, o, o,10, o,10, o, o, o,11, o,10, o, o, o,10, o,10, o, o, o, o, o,10, o, o, o,10]
85 : 10.90 [10,10,11,10, o,12,11,10,10, o,10,12,10,10, o,11, o,10,10, o,10,11,11,10, o,10,11,11,10, o,10,10,10, o, o,10,11,10,10, o,10,10]Para un Rectángulo de Fibonacci obviamente tenemos $f(F_{k+1},F_k)\le k$ y parece sencillo demostrar que este límite es agudo. Pero, ¿es esto realmente trivial?
Parece que bajo las restricciones anteriores sobre $m$ y $n$ los valores de $f(m,n)$ están muy cerca de $g(m)$ Más concretamente $$\boxed{\lfloor g(m)\rfloor -1\le f(m,n)\le \lceil g(m)\rceil +1}.$$ ¿Es posible que en los mosaicos mínimos se produzcan con frecuencia patrones como los de un "mosaico rectangular de Fibonacci"?
Tenga en cuenta que ya se sabe o al menos es plausible por el artículo citado en el primer hilo que $f(m,n)\sim g(m)$ .
¿Qué pasa con $f(m,n)$ si los lados del rectángulo no son coprimos? Obviamente $f(km,kn)\le f(m,n)$ para $k\in \mathbb N$ . En el rango de la tabla, hay igualdad en todas partes.
¿Se sabe algo sobre la conjetura $f(km,kn)= f(m,n)$ ?
Además, ¿existe un mosaico mínimo de un $km\times kn$ rectángulo tal que no todos los lados del cuadrado son múltiplos de $k$ ?
Para $m> n$ Llamemos a $m\times n$ rectángulo reducible si $f(m,n)=f(n,m-n)+1$ . Esto significa que hay un mosaico mínimo tal que el cuadrado más grande tiene lado $n$ . (Es posible que también existan otros tilings mínimos, aunque lo dudo bastante).
Utilizando el mismo orden que el anterior, he mantenido en fila $m$ sólo aquellos $n$ para los que el $m\times n$ rectángulo es reducible y poner "o" donde no lo es:
3 : [ 2]
4 : [ 3]
5 : [ 4, 3]
6 : [ o, 4]
7 : [ o, 5, 4]
8 : [ o, 6, 5]
9 : [ o, 7, 6, 5]
10 : [ o, 8, 7, 6]
11 : [ o, 9, 8, 7, 6]
12 : [ o, o, 9, 8, 7]
13 : [ o, o,10, 9, 8, 7]
14 : [ o, o,11,10, 9, 8]
15 : [ o, o,12,11,10, 9, 8]
16 : [ o, o,13,12,11,10, 9]
17 : [ o, o, o,13,12,11,10, 9]
18 : [ o, o, o,14,13,12,11,10]
19 : [ o, o, o, o, o,13,12,11,10]
20 : [ o, o, o,16,15,14,13,12,11]
21 : [ o, o, o,17,16,15,14,13,12,11]
22 : [ o, o, o,18,17,16, o,14,13,12]
23 : [ o, o, o,19,18,17,16, o,14,13,12]
24 : [ o, o, o, o,19,18,17,16,15,14,13]
25 : [ o, o, o, o,20,19,18,17,16,15,14,13]
26 : [ o, o, o, o, o,20,19,18,17,16,15,14]
27 : [ o, o, o, o, o,21,20,19,18,17,16,15,14]
28 : [ o, o, o, o,23,22,21,20,19,18,17,16, o]
29 : [ o, o, o, o,24,23, o,21,20,19,18,17,16,15]
30 : [ o, o, o, o, o,24,23,22,21,20,19,18,17,16]
31 : [ o, o, o, o, o, o, o, o, o,21,20,19,18,17,16]
32 : [ o, o, o, o, o,26,25,24,23,22,21,20,19,18,17]
33 : [ o, o, o, o, o,27, o,25,24,23,22,21,20,19,18,17]
34 : [ o, o, o, o, o, o,27,26, o,24,23,22,21,20,19,18]
35 : [ o, o, o, o, o, o,28,27, o,25,24,23,22,21,20,19,18]
36 : [ o, o, o, o, o, o, o,28,27,26,25,24,23,22,21,20,19]
37 : [ o, o, o, o, o,31, o, o,28,27,26,25,24,23, o,21,20,19]
38 : [ o, o, o, o, o, o, o, o,29, o,27,26,25,24,23,22,21,20]
39 : [ o, o, o, o, o, o,32, o,30,29,28,27,26,25,24,23,22,21, o]
40 : [ o, o, o, o, o, o, o,32, o,30, o,28,27,26,25,24,23,22,21]
41 : [ o, o, o, o, o, o, o, o, o,31,30,29, o,27,26,25,24,23,22,21]
42 : [ o, o, o, o, o, o, o,34,33,32,31,30,29,28,27,26,25,24,23,22]
43 : [ o, o, o, o, o, o, o,35, o, o,32,31, o,29,28,27,26,25, o,23,22]
44 : [ o, o, o, o, o, o, o,36, o,34,33,32,31, o,29,28,27,26,25,24,23]
45 : [ o, o, o, o, o, o, o, o,36,35, o,33,32,31,30,29,28,27,26,25,24,23]
46 : [ o, o, o, o, o, o, o,38, o,36,35,34,33,32,31, o,29,28,27,26,25,24]
47 : [ o, o, o, o, o, o, o, o, o, o, o, o,34,33,32,31, o,29,28,27,26,25,24]
48 : [ o, o, o, o, o, o, o, o,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25]
49 : [ o, o, o, o, o, o, o, o, o, o, o, o, o,35,34,33,32,31,30,29,28,27,26,25]
50 : [ o, o, o, o, o, o, o, o, o,40, o,38,37,36,35,34, o,32,31,30,29,28,27,26]
51 : [ o, o, o, o, o, o, o, o, o,41,40,39, o,37,36,35,34,33,32,31,30,29,28,27,26]
52 : [ o, o, o, o, o, o, o, o, o, o, o,40,39,38, o,36,35,34,33,32,31,30,29,28,27]
53 : [ o, o, o, o, o, o, o, o, o,43, o, o,40, o,38,37,36,35,34,33,32,31, o,29,28,27]
54 : [ o, o, o, o, o, o, o, o, o, o, o,42, o,40,39,38,37,36,35,34,33,32,31,30, o,28]
55 : [ o, o, o, o, o, o, o, o, o,45,44, o, o, o,40, o, o,37, o,35,34,33,32,31,30,29,28]
56 : [ o, o, o, o, o, o, o, o, o,46, o,44,43,42,41,40,39,38,37,36,35,34,33,32,31, o,29]
57 : [ o, o, o, o, o, o, o, o, o, o, o, o, o,43, o,41,40,39,38,37,36,35, o,33,32,31,30,29]
58 : [ o, o, o, o, o, o, o, o, o,48,47,46,45, o,43,42,41,40,39,38,37,36,35,34, o,32, o,30]
59 : [ o, o, o, o, o, o, o, o, o, o, o, o, o,45, o, o, o,41,40, o, o,37,36,35,34,33,32,31,30]
60 : [ o, o, o, o, o, o, o, o, o, o, o,48,47,46,45,44,43,42, o,40,39,38,37,36,35,34,33,32,31]
61 : [ o, o, o, o, o, o, o, o, o, o, o,49, o,47,46, o, o, o, o,41,40,39, o,37,36,35,34,33,32,31]
62 : [ o, o, o, o, o, o, o, o, o, o, o, o,49, o,47, o,45, o,43,42,41,40,39,38,37,36,35,34,33,32]
63 : [ o, o, o, o, o, o, o, o, o, o, o,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32]
64 : [ o, o, o, o, o, o, o, o, o, o, o,52, o,50,49,48, o,46,45,44,43,42,41,40,39,38,37,36,35,34,33]
65 : [ o, o, o, o, o, o, o, o, o, o, o, o,52, o,50,49,48,47,46,45,44,43, o,41,40,39,38,37,36,35,34,33]
66 : [ o, o, o, o, o, o, o, o, o, o, o,54, o, o,51,50, o,48,47,46, o,44,43,42,41,40,39,38,37,36,35,34]
67 : [ o, o, o, o, o, o, o, o, o, o, o,55, o, o, o,51, o,49,48,47,46,45,44,43, o,41,40,39,38,37, o, o,34]
68 : [ o, o, o, o, o, o, o, o, o, o, o, o, o,54, o,52,51, o,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35]
69 : [ o, o, o, o, o, o, o, o, o, o, o,57, o,55,54,53,52,51,50,49,48,47,46, o,44, o,42, o,40,39,38,37,36,35]
70 : [ o, o, o, o, o, o, o, o, o, o, o, o, o,56,55,54, o, o,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36]
71 : [ o, o, o, o, o, o, o, o, o, o, o,59, o, o, o,55,54, o,52, o,50,49,48,47,46,45,44,43, o,41,40,39,38,37,36]
72 : [ o, o, o, o, o, o, o, o, o, o, o, o, o, o,57,56, o,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37]
73 : [ o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o,55,54,53,52, o,50,49,48,47,46,45,44,43, o,41,40,39,38,37]
74 : [ o, o, o, o, o, o, o, o, o, o, o,62, o, o,59, o, o,56, o,54,53,52, o,50,49,48,47,46,45, o,43,42,41,40,39,38]
75 : [ o, o, o, o, o, o, o, o, o, o, o, o, o, o,60, o, o,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38]
76 : [ o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o,58,57, o,55,54, o,52,51,50,49,48,47,46,45,44,43,42,41,40,39]
77 : [ o, o, o, o, o, o, o, o, o, o, o, o, o,63, o, o,60, o, o,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39]
78 : [ o, o, o, o, o, o, o, o, o, o, o, o, o,64, o, o,61,60,59,58,57,56,55,54, o,52,51,50,49,48,47,46,45,44,43,42,41, o]
79 : [ o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o,61, o, o, o, o, o,55, o,53, o,51,50,49,48,47,46,45,44,43,42,41,40]
80 : [ o, o, o, o, o, o, o, o, o, o, o, o, o, o,65,64, o, o, o,60, o, o, o,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41]
81 : [ o, o, o, o, o, o, o, o, o, o, o, o, o,67, o, o, o,63, o, o,60,59, o,57,56,55,54, o,52,51,50,49,48,47,46,45,44,43,42,41]
82 : [ o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o,63,62,61,60, o,58, o, o,55,54, o,52, o,50,49,48,47,46,45,44,43,42]
83 : [ o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o,59, o,57,56,55, o,53,52,51,50,49,48,47,46,45,44,43,42]
84 : [ o, o, o, o, o, o, o, o, o, o, o, o, o, o,69,68, o,66, o,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46, o,44,43]
85 : [ o, o, o, o, o, o, o, o, o, o, o, o, o, o, o, o,68, o, o,65, o,63,62,61,60,59, o,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43]La tendencia general es clara, pero la situación global parece bastante irregular. Para algunos valores de $m$ Hay muchos más "agujeros" que para otros. ¿Alguna idea de por qué?
