Today, I just came by Haidian Book City as usual at Haidian dist in Beijing. and I found the problem hung on the wall nearby. It is very interesting and I want to share it.

0x00 Problem

Here is one picture of it.

27-problem_at_haidian_book_city

A translation of that:

Solve this problem, then it’s your domain: {3, 13, 1113, 3113,…, the 8th number}.angelcrunch.com
(the QR code leads to the below link) www.angelcrunch.com/jiemi

Once you finish it, you will get the second as below:

Guess a television series by the following numbers, and you will get an interview. 3113112211322112 / 311311

0x01 Solution

Yes, as you may guess, it is one look-and-say sequence(sequence A006715 in OEIS.

In the sewuence, each member is genrated from the previous menber by "reading" off the digits in it, counting rhe number of digits in groups of the same digit. For example:

  • 3 is reading off as “one 3” or 13.
  • 13 is reading off as “one 1 one 3” or 1113.
  • 1113 is reading off as “three 1s, then one 3” or 3113.
  • and so on.

If we start with any digit d from 0 to 9 then d will remain indefinitely as the last digit of the sequence. For d different from 1, the sequence starts as follows:

d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d, …

As example in the following table.

d Sloane sequence
1 A005150 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...
2 A006751 2, 12, 1112, 3112, 132112, 1113122112, 311311222112, ...
3 A006715 3, 13, 1113, 3113, 132113, 1113122113, 311311222113, ...

Here, d equals 3.

So the first answer is 13211321322113.

For the second one, you need to know more about the sequence. John Conway studied this sequence and found that the 8th member and every member after it in the sequence is made up of one or more of 92 “basic” non-interacting subsequences. The 92 basic subsequence shows in the following table(from here.

The fouth column in the table says what other subsequences the given subsequence evolves into. He also show that the number of the digits in each member of the sequence grows a constant from one member to the next. If Ln is the number of the digits in the nth member in the sequence, then Ln+1/Ln to a limitation when n to infinity. It is 1.303577269… , which we call it as Conway Constant.

No. Subsequence Length Evolves Into
1 1112 4 (63)
2 1112133 7 (64)(62)
3 111213322112 12 (65)
4 111213322113 12 (66)
5 1113 4 (68)
6 11131 5 (69)
7 111311222112 12 (84)(55)
8 111312 6 (70)
9 11131221 8 (71)
10 1113122112 10 (76)
11 1113122113 10 (77)
12 11131221131112 14 (82)
13 111312211312 12 (78)
14 11131221131211 14 (79)
15 111312211312113211 18 (80)
16 111312211312113221133211322112211213322112 42 (81)(29)(91)
17 111312211312113221133211322112211213322113 42 (81)(29)(90)
18 11131221131211322113322112 26 (81)(30)
19 11131221133112 14 (75)(29)(92)
20 1113122113322113111221131221 28 (75)(32)
21 11131221222112 14 (72)
22 111312212221121123222112 24 (73)
23 111312212221121123222113 24 (74)
24 11132 5 (83)
25 1113222 7 (86)
26 1113222112 10 (87)
27 1113222113 10 (88)
28 11133112 8 (89)(92)
29 12 2 (1)
30 123222112 9 (3)
31 123222113 9 (4)
32 12322211331222113112211 23 (2)(61)(29)(85)
33 13 2 (5)
34 131112 6 (28)
35 13112221133211322112211213322112 32 (24)(33)(61)(29)(91)
36 13112221133211322112211213322113 32 (24)(33)(61)(29)(90)
37 13122112 8 (7)
38 132 3 (8)
39 13211 5 (9)
40 132112 6 (10)
41 1321122112 10 (21)
42 132112211213322112 18 (22)
43 132112211213322113 18 (23)
44 132113 6 (11)
45 1321131112 10 (19)
46 13211312 8 (12)
47 1321132 7 (13)
48 13211321 8 (14)
49 132113212221 12 (15)
50 13211321222113222112 20 (18)
51 1321132122211322212221121123222112 34 (16)
52 1321132122211322212221121123222113 34 (17)
53 13211322211312113211 20 (20)
54 1321133112 10 (6)(61)(29)(92)
55 1322112 7 (26)
56 1322113 7 (27)
57 13221133112 11 (25)(29)(92)
58 1322113312211 13 (25)(29)(67)
59 132211331222113112211 21 (25)(29)(85)
60 13221133122211332 17 (25)(29)(68)(61)(29)(89)
61 22 2 (61)
62 3 1 (33)
63 3112 4 (40)
64 3112112 7 (41)
65 31121123222112 14 (42)
66 31121123222113 14 (43)
67 3112221 7 (38)(39)
68 3113 4 (44)
69 311311 6 (48)
70 31131112 8 (54)
71 3113112211 10 (49)
72 3113112211322112 16 (50)
73 3113112211322112211213322112 28 (51)
74 3113112211322112211213322113 28 (52)
75 311311222 9 (47)(38)
76 311311222112 12 (47)(55)
77 311311222113 12 (47)(56)
78 3113112221131112 16 (47)(57)
79 311311222113111221 18 (47)(58)
80 311311222113111221131221 24 (47)(59)
81 31131122211311122113222 23 (47)(60)
82 3113112221133112 16 (47)(33)(61)(29)(92)
83 311312 6 (45)
84 31132 5 (46)
85 311322113212221 15 (53)
86 311332 6 (38)(29)(89)
87 3113322112 10 (38)(30)
88 3113322113 10 (38)(31)
89 312 3 (34)
90 312211322212221121123222113 27 (36)
91 312211322212221121123222122 27 (35)
92 32112 5 (37)

Those 92 subsequence is so basic that is constructs every member in the look-and-say sequence. Just like 92 elements. Here gives the periodic table of atoms associated with the look-and-say sequence as named by Conway(1987). As we can see, 3113112211322112 links to Br, and 311311 links to Ba.

Breaking Bad. That is the answer.

0x02 More

That is perfect from the begining to the end. Many thanks to the problem maker, and the screenwriters, also every excellent actors in Breaking Bad.