Chain cube puzzles go by a number of different names. I have three different ones.
What you have are twenty-seven small cubes joined by an
elastic thread running through all of them.
The thread either passes straight though the cube
or makes a right angled turn.
Because the join is flexible you can turn it at will.
Here is a picture of one unwound all the way.
The object is to fold the structure into a large 3x3x3 cube.
Here is the same cube partially folded.
And completely solved.
There are 11,487 puzzles of this type. So, clearly most of them haven't been manufactured.
Here is a table I made a long time ago. The first column is the number of unique ways of folding the chain into a cube (ignoring rotations and reflections). The columns labelled '2' through '11' categorize chains based on the number of straight cubies. Each entry is the number of chains with that many straights and that many unique solutions. The last column is the sum of that row. The last row is the sum of the columns. For example, the chain pictured has 9 straight cubies and a unique solution so is one of the 187 chains with 9 straights that have a unique solutions.
2 3 4 5 6 7 8 9 10 11
1 15 144 589 1053 1078 556 187 34 2 3658
2 26 145 502 862 770 325 65 9 2704
3 25 118 326 393 255 104 21 1242
4 1 14 98 242 340 203 89 14 1 1002
5 12 56 140 168 86 12 1 475
6 1 11 57 123 181 97 23 3 496
7 2 6 36 101 71 30 246
8 8 35 89 83 43 22 7 1 288
9 3 9 23 51 61 17 5 169
10 10 33 57 34 16 6 156
11 6 28 39 22 11 1 1 108
12 2 4 30 39 35 17 9 1 137
13 2 2 20 31 9 5 1 70
14 6 23 25 24 10 88
15 2 16 28 12 3 61
16 3 15 24 11 7 5 3 1 69
17 1 9 14 16 9 3 52
18 1 3 9 6 10 7 1 37
19 6 12 16 6 2 42
20 1 11 18 11 1 42
21 3 4 8 10 3 1 29
22 1 6 10 7 2 1 27
23 1 5 8 6 2 22
24 2 8 7 6 3 26
25 5 5 5 1 1 17
26 1 8 5 3 1 18
27 1 2 6 5 4 18
28 1 1 10 2 1 1 16
29 1 4 5 10
30 4 3 1 4 12
31 3 1 7 1 12
32 2 4 6 2 14
33 1 3 3 3 1 11
34 2 3 4 9
35 1 1 2 4
36 2 2
37 2 1 1 4
38 1 1 1 3
39 4 2 6
40 1 3 4
41 1 1 2
42 1 3 4
43 1 2 1 4
44 1 3 1 5
45 2 1 3
46 1 1 2
47 1 1 2 2 6
48 2 1 3
49 1 1
50 1 1 1 3
51 1 1 2
52 1 1 2
53 1 1 2
54 2 2
55 2 2
56 1 1 2
57 2 1 3
58 1 1
* 61 1 1 2
62 1 1 3 5
* 64 1 1
* 67 1 1
* 70 1 1 1 3
71 1 1
* 73 1 1
* 81 1 1
* 85 1 1
86 1 1
87 1 1
88 2 2 4
* 90 1 1 2
*104 1 1 2
*112 1 1
*115 1 1
*119 1 1
*123 1 1
*126 1 1
*142 1 1
24 235 1037 2563 3444 2674 1159 303 46 2 11487