Snake Cubes

Snake Cubes, also called chain cubes or elastic cubes, are puzzles consisting of 27 cubes connected together, usually by a piece of elastic. Every cube (except for the ones at either end) has two faces with a hole in their centre through which the elastic runs. A cube can have holes in opposite faces, so that the elastic runs straight through, or in adjacent faces, so that the elastic makes a 90 degree turn through the cube. The aim of the puzzle is to fold it into the shape of a 3×3×3 cube.

There are many versions of this puzzle. Some are made of wood, others of plastic, but more interesting is that different versions have different arrangements of 'straight' and 'bend' cubes on the string. Some of these arrangements are much easier than others. In general, those with more straights are easier because there are fewer degrees of freedom. On the other hand, some versions may have multiple solutions making it easier, too. The most difficult ones are those with unique solutions and not too many straights.

This puzzle should not be confused with the Kibble cube. That is also a string of cubes, but its cubes has slots that allow the elastic to change the faces it enters and exits. There are also versions of the snake cube where it is easy to make a cube, but where the aim is to make one with a particular colour pattern. I will not discuss those here.

One of the hardest versions of the snake cube is Kev's Kubes (version 9B), made by Trench Puzzles. It is made from wood, alternately coloured white and black. A plastic version called Cubra comes in 5 variations, each with a different colour. In order of difficulty, they are Mean Green, Bafflin' Blue, Twist yer 'ead Red, 'Orrible Orange, and Puzzlin' Purple. Almost all other versions of the snake cube have the same shape as the Cubra Blue.

If we denote the straights, corners, and ends with the letters S, C, and E, then the type of snake cube is easily given by a string of 27 letters. Here are the snake patterns some of the commercially available puzzles:

 ESCCCSCSCCCCCCCCCCCCCSCSCCE Kev's Kubes (9B) ESCSCSCCSCSCSCCCCSCCSCCSCSE Cubra Green ESCSCSCSCCCCSCSCCCSCCSCCCSE Cubra Blue ESCCCCCCCCCSCCCCCCCSCSCCCCE Cubra Red ESCSCCCCSCCCCCCCCSCCCSCCCSE Cubra Orange ECCCCSCSCCCCCCCCCCCSCSCCCCE Cubra Purple

The Cubra Red has 10 solutions, the Cubra Purple has 6, and all the rest have unique solutions (disregarding mirror images or rotations).

If your browser supports it, you can click on the link below to play with a Javascript version of the Snake Cube.

Analysis:

The string of cubes form a Hamiltonian path through a 3×3×3 grid graph. In other words, it is a path through the 27 points of a 3×3×3 grid, that visits each point exactly once.
I wrote a computer program that finds all such Hamiltonian paths, discarding any rotations or mirror images. The 51704 produced paths were then examined to see what string of straights and bends they form, which resulted in 11487 snakes. Of these, 3658 have unique solutions. The full results are listed in the tables below.

N
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b
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o
f

S
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n
s
Number of Straights
234567891011Total
1015144589105310785561873423658
202614550286277032565902704
302511832639325510421001242
4114982423402038914101002
5012561401688612100475
6111571231819723300496
7263610171300000246
8083589834322710288
939235161175000169
10010335734166000156
1106283922111100108
1224303935179100137
1322203195100070
140623252410000088
15021628123000061
16031524117531069
1719141693000052
181396107100037
1906121662000042
20011118111000042
213481031000029
221610721000027
23158620000022
24028763000026
25055511000017
26108531000018
27126540000018
281110211000016
29014500000010
30004314000012
31031710000012
32024620000014
33013331000011
3402304000009
3501120000004
3600020000002
3702110000004
3801110000003
3900420000006
4000013000004
4100101000002
4200103000004
4312001000004
4401031000005
4500021000003
4601100000002
4701122000006
4802001000003
4900010000001
5001110000003
5100110000002
5200101000002
5300110000002
5402000000002
5502000000002
5610100000002
5700201000003
5800100000001
6101100000002
6211030000005
6400100000001
6700010000001
7001011000003
7101000000001
7300100000001
8100100000001
8500010000001
8600100000001
8700100000001
8800220000004
9000110000002
10401100000002
11201000000001
11500010000001
11900100000001
12301000000001
12600100000001
14200100000001
Total   24   235   1037   2563   3444   2674   1159   303   46   2   11487

The following is a similar table, but counting only those snake patterns that are palindromic, i.e. sequences of straights and bends which look the same when reversed, such as the Cubra Purple.

234567891011   Total
10054610211029
200113310009
300031012007
411221200009
500002200004
600101000002
700030000003
901000100002
1001000000001
1100100001002
1310010000002
1700100000001
1900010000001
2110000000001
2201000000001
2500100000001
3101000000001
4500010000001
Total   3   5   12   16   14   18   4   4   1   0   77

As you can see, it is impossible to form a cube using a snake with only bends in it. It is possible to use longer snakes without straights to form a 5×5×5 cube, or a 3×5×7 block. Blocks with one or more even dimensions, such as a 2×2×2 cube, tend to be too easy.

I'll just list some of the outliers in the above tables.

• The two snakes with 11 straights:
ESCSCSCCSCSCSCCSCSCSCCSCCSE
ESCCSCCSCSCSCSCSCSCSCCSCSCE
• The 15 snakes with 3 straights and unique solutions:
ECCCCCCCCSCSCCCSCCCCCCCCCCE     ECCCCCCCSCCSCSCCCCCCCCCCCCE
ECCCCCCSCCCSCSCCCCCCCCCCCCE     ECCCCSCCCCCCCCCSCCSCCCCCCCE
ECCCCSCCCCCCCSCCCSCCCCCCCCE     ECCCCSCCCCCSCCCSCCCCCCCCCCE
ECCCCSCCCSCCCCCCCSCCCCCCCCE     ECCCCSCCCSCCCSCCCCCCCCCCCCE
ECCCCSCCSCCCCCCCCSCCCCCCCCE     ECCSCCCSCCCCCSCCCCCCCCCCCCE
ECSCCSCCCCCCCCCCCCCSCCCCCCE     ESCCCCCCCCSCCCCCCCCCCCCCCSE
ESCCCCCCCSCCCSCCCCCCCCCCCCE     ESCCCSCCCCCCCCCCCCCCCSCCCCE
ESCCCSCCCCCCCSCCCCCCCCCCCCE
• The palindromic snakes with unique solutions (followed by the number of straights):
ECCCCSCCCCCSCCCSCCCCCSCCCCE   4
ESCCCCCCCCCSCCCSCCCCCCCCCSE   4
ESCCCCCCCSCCCCCCCSCCCCCCCSE   4
ESCCSCCCCCCCCCCCCCCCCCSCCSE   4
ECCCCSCCSCCCCCCCCCSCCSCCCCE   4
ECCCCCCCSCCSCSCSCCSCCCCCCCE   5
ESCCSCCCCCCCCSCCCCCCCCSCCSE   5
ECCCCSCSCCCCCSCCCCCSCSCCCCE   5
ECCCCCSCCSCCCSCCCSCCSCCCCCE   5
ECCSCSCCSCCCCCCCCCSCCSCSCCE   6
ESCCSCCCCCCSCCCSCCCCCCSCCSE   6
ESCCSCCCSCCCCCCCCCSCCCSCCSE   6
ECCCCSCSCCSCCCCCSCCSCSCCCCE   6
ECCSCSCSCCCCCCCCCCCSCSCSCCE   6
ECCSCSCCCCCSCCCSCCCCCSCSCCE   6
ESCCCCCCSCCSCSCSCCSCCCCCCSE   7
ECCSCCCCCSCSCSCSCSCCCCCSCCE   7
ESCSCCCCSCCCCSCCCCSCCCCSCSE   7
ESCCCCSCCSCCCSCCCSCCSCCCCSE   7
ECSCCSCSCCCCCSCCCCCSCSCCSCE   7
ECSCCCCSCSCCCSCCCSCSCCCCSCE   7
ECCCCSCSCSCCCSCCCSCSCSCCCCE   7
ESCCSCCCCCCSCSCSCCCCCCSCCSE   7
ECSCCCCCCSCSCSCSCSCCCCCCSCE   7
ECCSCSCSCCCCCSCCCCCSCSCSCCE   7
ESCSCSCCCCSCCCCCSCCCCSCSCSE   8
ESCCSCSCCSCCCCCCCSCCSCSCCSE   8
ESCSCSCCSCCCCSCCCCSCCSCSCSE   9
ESCCSCCSCSCCSCSCCSCSCCSCCSE   10
• Some of the snakes with most solutions:
ESCCSCCSCSCCCCCCCCCCCCCCCCE   90
ECCCCCCSCSCCSCCSCSCCCCCCCCE   90
ESCSCCCCCSCCCCCCCCCCCCCCCCE   104
ECCSCSCSCSCCCCCCCCCCCCCCCCE   104
ECCCCSCSCSCCCCCCCCCCCCCCCCE   112
ESCSCSCSCSCCCCCCCCCCCCCCCCE   115
ESCSCSCSCCCCCCCCCCCCCCCCCCE   119
ESCSCSCCCCCCCCCCCCCCCCCCCCE   123
ECCCCSCSCSCSCCCCCCCCCCCCCCE   126
ECCCCCCSCSCSCSCCCCCCCCCCCCE   142
• Some of the palindromic snakes with most solutions:
ESCCCCCCCCCCCCCCCCCCCCCCCSE   21
ESCCCCCCCCCCCSCCCCCCCCCCCSE   22
ECCCCCCSCSCCCCCCCSCSCCCCCCE   25
ECCCCCCCCCCSCSCSCCCCCCCCCCE   31
ECCCCCCCCSCSCSCSCSCCCCCCCCE   45

General hints:

If the cubes of the snake alternate in colour, it is clear that one colour, e.g. white, has one more cube than the other. The finished cube will also have cubes alternating in colour, and it follows that the white cubes form the corners and face centres.
The two ends of the cube must therefore also be at corners and/or centres.

If the snake has several straights close to each other, try to solve that section first.

If the snake has a straight that is white (same colour as the snake's ends) then it must lie at a face centre. It is often useful to start solving from there.

Solutions:

The solutions will also be given using the letters F, L, U, B, R, D standing for the six directions in space where the next cube might be, viz. Front, Left, Up, Back, Right, Down respectively.

Kev's Kubes (9B):

As mentioned before, this is one of the hardest snake cubes. One contributing factor is that the ends of the snake both lie in face centres, not at corners.

 RR F D LL BB R U L U R F L F D R U R BB DD F L

Cubra Green:

 RR BB LL U RR FF LL U R B DD L UU B RR FF

Cubra Blue:

 RR BB LL UU R D R FF LL U B DD R UU F R BB

Cubra Red:

 RR B U F U L D L U BB R D F U R B DD LL U F D R RR B U F U L D L U BB D F R D L B RR UU F L B D RR B U F U L D L U BB D R U F R B DD LL F U R D RR B U F U L D L U BB D R U F R B DD LL F R U L RR B U F U B L F D BB L D F R B R UU LL F D F U RR B U F U B L B R DD L U F D L B UU FF R D L B RR B U F L U R B L DD L B U R D R UU LL F D F U RR B U F L B U L B DD R F L U F U RR BB L D R D RR B U F L B D L B UU R F L D F U RR BB D L D R RR B L B R U L U R FF D B L U F L BB DD F U F R

Cubra Orange:

 RR BB L U R FF L U L D B U B DD F R UU B R FF

Cubra Purple:

 R B L B UU RR F L D L U F D R U R DD BB L U R F R B L B UU RR F L D B D R U F D F UU LL B D F R R B L B RR UU F L B L D R F R D F UU LL B D F R R B L B RR UU F L B D L U F D F U RR DD B U L F R B R F UU LL B R D B D R U F U B LL DD F U F R R B R F UU LL B R B L D R F R U B DD LL F U F R