Tantrix rock.

Tantrix ROCK.
Version of 22 November 2009.
Dave Barber's other pages.

The Tantrix ROCK is a three-dimensional puzzle where the player tries to assemble eight hexagonal tiles and six square tiles into a truncated octahedron. Here are several views of that shape:

— Figure one —

The challenge of the puzzle is that each tile has markings which must match where one tile adjoins another. More specifically, the manufacturer paints paths, straight or curved, on each tile with each path connecting the middle of one edge with another. Consequently, a hexagonal tile has three paths and a square tile has two; paths sometimes cross each other. By convention, all paths on a tile are different colors, typically yellow, red and blue. A green stripe is also found on some non-ROCK versions of Tantrix, which usually involve tessallating hexagons on a plane.

Rather than colors, this report uses single, double or triple stripes in deference to people who have difficulty distinguishing colors; there is also an advantage should the images be printed or viewed on a monochrome device. The logic of the puzzle is the same.

An extended discussion of Tantrix is found at Jaap.

Tiles can exhibit any of the configurations in table one; under each is written a label for reference.

Table one

6:0
6:2
6:5
6:8
6:B
6:E
4:0
4:3

6:1
6:3
6:6
6:9
6:C
6:F
4:1
4:4

6:4
6:7
6:A
6:D
4:2
4:5

Table one
6:0	6:2	6:5	6:8	6:B	6:E	4:0	4:3
6:1	6:3	6:6	6:9	6:C	6:F	4:1	4:4
	6:4	6:7	6:A	6:D		4:2	4:5

The factory supplies only eight hexagonal tiles with the ROCK, and they have yellow, blue or red stripes. Under the following equivalences…

yellow ↔ single stripe
blue ↔ double stripe
red ↔ triple stripe

…the factory hexagons correspond to 6:0, 6:2, 6:3, 6:5, 6:6, 6:9, 6:B and 6:C of the chart. With them, five solutions are possible. Not apparent is the rationale for including some tiles and omitting others, as any of them might be used in a solution.

Our analysis will assume that all sixteen hexagons are available, although only eight can be used at a time; then 476 solutions will be generated.

The factory usually omits tiles 6:E and 6:F, containing only straight lines, from those Tantrix sets intended for plane tessellation. This is because those tiles have only three distinct rotations rather than six, and this lack is felt to give players insufficient flexibility. If Tantrix ROCK is similarly constrained, only 208 solutions will be possible.

All six square tiles are required in every solution.

Here are several views of one solution:

— Figure two —

Although such pictorial images are helpful, they cannot show all tiles at once. An alternative is to arrange tiles as though the truncated octahedron is cut apart along certain edges, forming a net:

— Figure three —

Such a cut-apart drawing requires about 11 kilobytes, which if produced for each of the 476 solutions would require exorbitant amounts of memory. Our presentation of the solutions instead uses a separate image for each tile, at some loss of clarity:

— Figure four —

Because images are reused from one solution to the next, memory demands and download time are greatly reduced. A lengthy table contains all solutions.

Although not chosen for this report, a more compact representation is possible, as in figure five. Because placement of the squares is trivial once the hexagons are established, the squares could be omitted entirely.

— Figure five —

Different combinations of hexagons offer different numbers of solutions. In table two, the "6:" has been omitted for clarity.

Table two
Each has 12 solutions:

• 0 1 2 3 4 5 8 E
• 0 1 2 3 4 5 8 F
• 0 1 2 3 4 6 9 E
• 0 1 2 3 4 6 9 F
• 0 1 2 3 4 7 A E
• 0 1 2 3 4 7 A F Each has 5 solutions:

• 0 2 3 5 6 9 B C
• 0 2 3 5 7 A B C
• 0 2 4 5 7 8 B D
• 0 2 4 6 7 9 B D
• 0 3 4 5 6 8 C D
• 0 3 4 6 7 A C D
• 1 2 3 6 8 9 B C
• 1 2 3 7 8 A B C
• 1 2 4 5 8 A B D
• 1 2 4 6 9 A B D
• 1 3 4 5 8 9 C D
• 1 3 4 7 9 A C D Each has 4 solutions:

• 0 1 2 3 4 5 6 7
• 0 1 2 3 4 8 9 A
• 0 1 5 6 7 B C D
• 0 1 8 9 A B C D
• 0 2 3 6 7 B C F
• 0 2 4 5 6 B D F
• 0 3 4 5 7 C D F
• 1 2 3 9 A B C E
• 1 2 4 8 9 B D E
• 1 3 4 8 A C D E Each has 3 solutions:

• 0 2 5 6 7 8 D F
• 0 2 5 6 7 A C F
• 0 3 5 6 7 8 D F
• 0 3 5 6 7 9 B F
• 0 4 5 6 7 9 B F
• 0 4 5 6 7 A C F
• 1 2 5 8 9 A D E
• 1 2 7 8 9 A C E
• 1 3 5 8 9 A D E
• 1 3 6 8 9 A B E
• 1 4 6 8 9 A B E
• 1 4 7 8 9 A C E

Table two
Each has 12 solutions: • 0 1 2 3 4 5 8 E • 0 1 2 3 4 5 8 F • 0 1 2 3 4 6 9 E • 0 1 2 3 4 6 9 F • 0 1 2 3 4 7 A E • 0 1 2 3 4 7 A F	Each has 5 solutions: • 0 2 3 5 6 9 B C • 0 2 3 5 7 A B C • 0 2 4 5 7 8 B D • 0 2 4 6 7 9 B D • 0 3 4 5 6 8 C D • 0 3 4 6 7 A C D • 1 2 3 6 8 9 B C • 1 2 3 7 8 A B C • 1 2 4 5 8 A B D • 1 2 4 6 9 A B D • 1 3 4 5 8 9 C D • 1 3 4 7 9 A C D	Each has 4 solutions: • 0 1 2 3 4 5 6 7 • 0 1 2 3 4 8 9 A • 0 1 5 6 7 B C D • 0 1 8 9 A B C D • 0 2 3 6 7 B C F • 0 2 4 5 6 B D F • 0 3 4 5 7 C D F • 1 2 3 9 A B C E • 1 2 4 8 9 B D E • 1 3 4 8 A C D E	Each has 3 solutions: • 0 2 5 6 7 8 D F • 0 2 5 6 7 A C F • 0 3 5 6 7 8 D F • 0 3 5 6 7 9 B F • 0 4 5 6 7 9 B F • 0 4 5 6 7 A C F • 1 2 5 8 9 A D E • 1 2 7 8 9 A C E • 1 3 5 8 9 A D E • 1 3 6 8 9 A B E • 1 4 6 8 9 A B E • 1 4 7 8 9 A C E

Meanwhile, 68 combinations have two solutions, and 132 have one. This leaves 12630 combinations of hexagonal tiles that give no solution at all.

If all 16 hexagonal tiles are available, then once a solution is obtained, certain operations can yield more solutions. Details of twelve operations are in table three, but a few of them are enough to generate the others:

Exchange single-stripe paths with double-stripe (rc₁).
Exchange double-stripe paths with triple-stripe (rc₅).
Reflect (take a mirror image of) the whole figure (rt₀).

Table three shows which tiles are replaced by which others under each operation. Consider for instance exchanging singles and triples (rc₃) within an old solution to obtain a new solution. In the rc₃ column of the upper part of the table, the expression 6:A >> 6:5 means that if the tile 6:A was in the old solution, 6:5 will be installed in the new solution, in whatever location 6:A had been. Tile 6:A might ultimately be used elsewhere in the new solution, or be discarded. By the same token, 6:5 may or may not have appeared in the old solution.

The difference between the upper part (rc) and the lower part (rt) is that operations in the lower also involve reflecting the solution. To denote this difference, column headers use single-shaft arrows in the upper part, and double-shaft in the lower. In the operation names, r stands for rearrangement, c for cis, and t for trans.

In a fuller example, the hexagonal tiles of solution 283 — 6:0, 6:2, 6:5, 6:6, 6:7, 6:9, 6:A and 6:D — would under operation rt₄ become the hexagon tiles of solution 459 — 6:1, 6:4, 6:A, 6:8, 6:9, 6:5, 6:6 and 6:C respectively.

Although all square tiles are used in every solution, they are included in the tables to show how they move around. For example, tiles 4:0 and 4:1 trade places in rt₃.

Operations can be performed one after the other, but the sequence often makes a difference. For example, first exchanging single and double stripes (rc₁), and then exchanging double and triple stripes (rc₅), is equivalent to changing singles to triples, doubles to singles, and triples to doubles (rc₄). On the other hand, rc₅ followed by rc₁ gives rc₂ instead.

Table three
without
reflection rc₀
single → single
double → double
triple → triple rc₁
single → double
double → single
triple → triple rc₂
single → double
double → triple
triple → single rc₃
single → triple
double → double
triple → single rc₄
single → triple
double → single
triple → double rc₅
single → single
double → triple
triple → double
6:0 >> 6:0
6:1 >> 6:1

6:2 >> 6:2
6:3 >> 6:3
6:4 >> 6:4

6:5 >> 6:5
6:6 >> 6:6
6:7 >> 6:7
6:8 >> 6:8
6:9 >> 6:9
6:A >> 6:A

6:B >> 6:B
6:C >> 6:C
6:D >> 6:D

6:E >> 6:E
6:F >> 6:F 6:0 >> 6:1
6:1 >> 6:0

6:2 >> 6:4
6:3 >> 6:3
6:4 >> 6:2

6:5 >> 6:9
6:6 >> 6:8
6:7 >> 6:A
6:8 >> 6:6
6:9 >> 6:5
6:A >> 6:7

6:B >> 6:D
6:C >> 6:C
6:D >> 6:B

6:E >> 6:F
6:F >> 6:E 6:0 >> 6:0
6:1 >> 6:1

6:2 >> 6:3
6:3 >> 6:4
6:4 >> 6:2

6:5 >> 6:6
6:6 >> 6:7
6:7 >> 6:5
6:8 >> 6:9
6:9 >> 6:A
6:A >> 6:8

6:B >> 6:C
6:C >> 6:D
6:D >> 6:B

6:E >> 6:E
6:F >> 6:F 6:0 >> 6:1
6:1 >> 6:0

6:2 >> 6:2
6:3 >> 6:4
6:4 >> 6:3

6:5 >> 6:A
6:6 >> 6:9
6:7 >> 6:8
6:8 >> 6:7
6:9 >> 6:6
6:A >> 6:5

6:B >> 6:B
6:C >> 6:D
6:D >> 6:C

6:E >> 6:F
6:F >> 6:E 6:0 >> 6:0
6:1 >> 6:1

6:2 >> 6:4
6:3 >> 6:2
6:4 >> 6:3

6:5 >> 6:7
6:6 >> 6:5
6:7 >> 6:6
6:8 >> 6:A
6:9 >> 6:8
6:A >> 6:9

6:B >> 6:D
6:C >> 6:B
6:D >> 6:C

6:E >> 6:E
6:F >> 6:F 6:0 >> 6:1
6:1 >> 6:0

6:2 >> 6:3
6:3 >> 6:2
6:4 >> 6:4

6:5 >> 6:8
6:6 >> 6:A
6:7 >> 6:9
6:8 >> 6:5
6:9 >> 6:7
6:A >> 6:6

6:B >> 6:C
6:C >> 6:B
6:D >> 6:D

6:E >> 6:F
6:F >> 6:E
4:0 >> 4:0
4:1 >> 4:1
4:2 >> 4:2

4:3 >> 4:3
4:4 >> 4:4
4:5 >> 4:5 4:0 >> 4:0
4:1 >> 4:2
4:2 >> 4:1

4:3 >> 4:3
4:4 >> 4:5
4:5 >> 4:4 4:0 >> 4:1
4:1 >> 4:2
4:2 >> 4:0

4:3 >> 4:4
4:4 >> 4:5
4:5 >> 4:3 4:0 >> 4:1
4:1 >> 4:0
4:2 >> 4:2

4:3 >> 4:4
4:4 >> 4:3
4:5 >> 4:5 4:0 >> 4:2
4:1 >> 4:0
4:2 >> 4:1

4:3 >> 4:5
4:4 >> 4:3
4:5 >> 4:4 4:0 >> 4:2
4:1 >> 4:1
4:2 >> 4:0

4:3 >> 4:5
4:4 >> 4:4
4:5 >> 4:3

with
reflection rt₀
single ⇒ single
double ⇒ double
triple ⇒ triple rt₁
single ⇒ double
double ⇒ single
triple ⇒ triple rt₂
single ⇒ double
double ⇒ triple
triple ⇒ single rt₃
single ⇒ triple
double ⇒ double
triple ⇒ single rt₄
single ⇒ triple
double ⇒ single
triple ⇒ double rt₅
single ⇒ single
double ⇒ triple
triple ⇒ double
6:0 >> 6:1
6:1 >> 6:0

6:2 >> 6:2
6:3 >> 6:3
6:4 >> 6:4

6:5 >> 6:8
6:6 >> 6:9
6:7 >> 6:A
6:8 >> 6:5
6:9 >> 6:6
6:A >> 6:7

6:B >> 6:B
6:C >> 6:C
6:D >> 6:D

6:E >> 6:F
6:F >> 6:E 6:0 >> 6:0
6:1 >> 6:1

6:2 >> 6:4
6:3 >> 6:3
6:4 >> 6:2

6:5 >> 6:6
6:6 >> 6:5
6:7 >> 6:7
6:8 >> 6:9
6:9 >> 6:8
6:A >> 6:A

6:B >> 6:D
6:C >> 6:C
6:D >> 6:B

6:E >> 6:E
6:F >> 6:F 6:0 >> 6:1
6:1 >> 6:0

6:2 >> 6:3
6:3 >> 6:4
6:4 >> 6:2

6:5 >> 6:9
6:6 >> 6:A
6:7 >> 6:8
6:8 >> 6:6
6:6 >> 6:7
6:A >> 6:5

6:B >> 6:C
6:C >> 6:D
6:D >> 6:B

6:E >> 6:F
6:F >> 6:E 6:0 >> 6:0
6:1 >> 6:1

6:2 >> 6:2
6:3 >> 6:4
6:4 >> 6:3

6:5 >> 6:7
6:6 >> 6:6
6:7 >> 6:5
6:8 >> 6:A
6:9 >> 6:9
6:A >> 6:8

6:B >> 6:B
6:C >> 6:D
6:D >> 6:C

6:E >> 6:E
6:F >> 6:F 6:0 >> 6:1
6:1 >> 6:0

6:2 >> 6:4
6:3 >> 6:2
6:4 >> 6:3

6:5 >> 6:A
6:6 >> 6:8
6:7 >> 6:9
6:8 >> 6:7
6:9 >> 6:5
6:A >> 6:6

6:B >> 6:D
6:C >> 6:B
6:D >> 6:C

6:E >> 6:F
6:F >> 6:E 6:0 >> 6:0
6:1 >> 6:1

6:2 >> 6:3
6:3 >> 6:2
6:4 >> 6:4

6:5 >> 6:5
6:6 >> 6:7
6:7 >> 6:6
6:8 >> 6:8
6:9 >> 6:A
6:A >> 6:9

6:B >> 6:C
6:C >> 6:B
6:D >> 6:D

6:E >> 6:E
6:F >> 6:F
4:0 >> 4:0
4:1 >> 4:1
4:2 >> 4:2

4:3 >> 4:3
4:4 >> 4:4
4:5 >> 4:5 4:0 >> 4:0
4:1 >> 4:2
4:2 >> 4:1

4:3 >> 4:3
4:4 >> 4:5
4:5 >> 4:4 4:0 >> 4:1
4:1 >> 4:2
4:2 >> 4:0

4:3 >> 4:4
4:4 >> 4:5
4:5 >> 4:3 4:0 >> 4:1
4:1 >> 4:0
4:2 >> 4:2

4:3 >> 4:4
4:4 >> 4:3
4:5 >> 4:5 4:0 >> 4:2
4:1 >> 4:0
4:2 >> 4:1

4:3 >> 4:5
4:4 >> 4:3
4:5 >> 4:4 4:0 >> 4:2
4:1 >> 4:1
4:2 >> 4:0

4:3 >> 4:5
4:4 >> 4:4
4:5 >> 4:3

Table three
without reflection	rc₀ single → single double → double triple → triple	rc₁ single → double double → single triple → triple	rc₂ single → double double → triple triple → single	rc₃ single → triple double → double triple → single	rc₄ single → triple double → single triple → double	rc₅ single → single double → triple triple → double
6:0 >> 6:0 6:1 >> 6:1 6:2 >> 6:2 6:3 >> 6:3 6:4 >> 6:4 6:5 >> 6:5 6:6 >> 6:6 6:7 >> 6:7 6:8 >> 6:8 6:9 >> 6:9 6:A >> 6:A 6:B >> 6:B 6:C >> 6:C 6:D >> 6:D 6:E >> 6:E 6:F >> 6:F	6:0 >> 6:1 6:1 >> 6:0 6:2 >> 6:4 6:3 >> 6:3 6:4 >> 6:2 6:5 >> 6:9 6:6 >> 6:8 6:7 >> 6:A 6:8 >> 6:6 6:9 >> 6:5 6:A >> 6:7 6:B >> 6:D 6:C >> 6:C 6:D >> 6:B 6:E >> 6:F 6:F >> 6:E	6:0 >> 6:0 6:1 >> 6:1 6:2 >> 6:3 6:3 >> 6:4 6:4 >> 6:2 6:5 >> 6:6 6:6 >> 6:7 6:7 >> 6:5 6:8 >> 6:9 6:9 >> 6:A 6:A >> 6:8 6:B >> 6:C 6:C >> 6:D 6:D >> 6:B 6:E >> 6:E 6:F >> 6:F	6:0 >> 6:1 6:1 >> 6:0 6:2 >> 6:2 6:3 >> 6:4 6:4 >> 6:3 6:5 >> 6:A 6:6 >> 6:9 6:7 >> 6:8 6:8 >> 6:7 6:9 >> 6:6 6:A >> 6:5 6:B >> 6:B 6:C >> 6:D 6:D >> 6:C 6:E >> 6:F 6:F >> 6:E	6:0 >> 6:0 6:1 >> 6:1 6:2 >> 6:4 6:3 >> 6:2 6:4 >> 6:3 6:5 >> 6:7 6:6 >> 6:5 6:7 >> 6:6 6:8 >> 6:A 6:9 >> 6:8 6:A >> 6:9 6:B >> 6:D 6:C >> 6:B 6:D >> 6:C 6:E >> 6:E 6:F >> 6:F	6:0 >> 6:1 6:1 >> 6:0 6:2 >> 6:3 6:3 >> 6:2 6:4 >> 6:4 6:5 >> 6:8 6:6 >> 6:A 6:7 >> 6:9 6:8 >> 6:5 6:9 >> 6:7 6:A >> 6:6 6:B >> 6:C 6:C >> 6:B 6:D >> 6:D 6:E >> 6:F 6:F >> 6:E
4:0 >> 4:0 4:1 >> 4:1 4:2 >> 4:2 4:3 >> 4:3 4:4 >> 4:4 4:5 >> 4:5	4:0 >> 4:0 4:1 >> 4:2 4:2 >> 4:1 4:3 >> 4:3 4:4 >> 4:5 4:5 >> 4:4	4:0 >> 4:1 4:1 >> 4:2 4:2 >> 4:0 4:3 >> 4:4 4:4 >> 4:5 4:5 >> 4:3	4:0 >> 4:1 4:1 >> 4:0 4:2 >> 4:2 4:3 >> 4:4 4:4 >> 4:3 4:5 >> 4:5	4:0 >> 4:2 4:1 >> 4:0 4:2 >> 4:1 4:3 >> 4:5 4:4 >> 4:3 4:5 >> 4:4	4:0 >> 4:2 4:1 >> 4:1 4:2 >> 4:0 4:3 >> 4:5 4:4 >> 4:4 4:5 >> 4:3

with reflection	rt₀ single ⇒ single double ⇒ double triple ⇒ triple	rt₁ single ⇒ double double ⇒ single triple ⇒ triple	rt₂ single ⇒ double double ⇒ triple triple ⇒ single	rt₃ single ⇒ triple double ⇒ double triple ⇒ single	rt₄ single ⇒ triple double ⇒ single triple ⇒ double	rt₅ single ⇒ single double ⇒ triple triple ⇒ double
6:0 >> 6:1 6:1 >> 6:0 6:2 >> 6:2 6:3 >> 6:3 6:4 >> 6:4 6:5 >> 6:8 6:6 >> 6:9 6:7 >> 6:A 6:8 >> 6:5 6:9 >> 6:6 6:A >> 6:7 6:B >> 6:B 6:C >> 6:C 6:D >> 6:D 6:E >> 6:F 6:F >> 6:E	6:0 >> 6:0 6:1 >> 6:1 6:2 >> 6:4 6:3 >> 6:3 6:4 >> 6:2 6:5 >> 6:6 6:6 >> 6:5 6:7 >> 6:7 6:8 >> 6:9 6:9 >> 6:8 6:A >> 6:A 6:B >> 6:D 6:C >> 6:C 6:D >> 6:B 6:E >> 6:E 6:F >> 6:F	6:0 >> 6:1 6:1 >> 6:0 6:2 >> 6:3 6:3 >> 6:4 6:4 >> 6:2 6:5 >> 6:9 6:6 >> 6:A 6:7 >> 6:8 6:8 >> 6:6 6:6 >> 6:7 6:A >> 6:5 6:B >> 6:C 6:C >> 6:D 6:D >> 6:B 6:E >> 6:F 6:F >> 6:E	6:0 >> 6:0 6:1 >> 6:1 6:2 >> 6:2 6:3 >> 6:4 6:4 >> 6:3 6:5 >> 6:7 6:6 >> 6:6 6:7 >> 6:5 6:8 >> 6:A 6:9 >> 6:9 6:A >> 6:8 6:B >> 6:B 6:C >> 6:D 6:D >> 6:C 6:E >> 6:E 6:F >> 6:F	6:0 >> 6:1 6:1 >> 6:0 6:2 >> 6:4 6:3 >> 6:2 6:4 >> 6:3 6:5 >> 6:A 6:6 >> 6:8 6:7 >> 6:9 6:8 >> 6:7 6:9 >> 6:5 6:A >> 6:6 6:B >> 6:D 6:C >> 6:B 6:D >> 6:C 6:E >> 6:F 6:F >> 6:E	6:0 >> 6:0 6:1 >> 6:1 6:2 >> 6:3 6:3 >> 6:2 6:4 >> 6:4 6:5 >> 6:5 6:6 >> 6:7 6:7 >> 6:6 6:8 >> 6:8 6:9 >> 6:A 6:A >> 6:9 6:B >> 6:C 6:C >> 6:B 6:D >> 6:D 6:E >> 6:E 6:F >> 6:F
4:0 >> 4:0 4:1 >> 4:1 4:2 >> 4:2 4:3 >> 4:3 4:4 >> 4:4 4:5 >> 4:5	4:0 >> 4:0 4:1 >> 4:2 4:2 >> 4:1 4:3 >> 4:3 4:4 >> 4:5 4:5 >> 4:4	4:0 >> 4:1 4:1 >> 4:2 4:2 >> 4:0 4:3 >> 4:4 4:4 >> 4:5 4:5 >> 4:3	4:0 >> 4:1 4:1 >> 4:0 4:2 >> 4:2 4:3 >> 4:4 4:4 >> 4:3 4:5 >> 4:5	4:0 >> 4:2 4:1 >> 4:0 4:2 >> 4:1 4:3 >> 4:5 4:4 >> 4:3 4:5 >> 4:4	4:0 >> 4:2 4:1 >> 4:1 4:2 >> 4:0 4:3 >> 4:5 4:4 >> 4:4 4:5 >> 4:3

These twelve operations form a symmetry group. Operation rc₀, which changes nothing, is the identity element.

Table four shows all the possibilities when one operation comes after another. For example, if rc₃ is followed by rt₂, the effect is the same as if rt₅ alone had been performed.

Table four
second operation
rc₀ rc₁ rc₂ rc₃ rc₄ rc₅ rt₀ rt₁ rt₂ rt₃ rt₄ rt₅
first
operation rc₀ rc₀ rc₁ rc₂ rc₃ rc₄ rc₅ rt₀ rt₁ rt₂ rt₃ rt₄ rt₅
rc₁ rc₁ rc₀ rc₃ rc₂ rc₅ rc₄ rt₁ rt₀ rt₃ rt₂ rt₅ rt₄
rc₂ rc₂ rc₅ rc₄ rc₁ rc₀ rc₃ rt₂ rt₅ rt₄ rt₁ rt₀ rt₃
rc₃ rc₃ rc₄ rc₅ rc₀ rc₁ rc₂ rt₃ rt₄ rt₅ rt₀ rt₁ rt₂
rc₄ rc₄ rc₃ rc₀ rc₅ rc₂ rc₁ rt₄ rt₃ rt₀ rt₅ rt₂ rt₁
rc₅ rc₅ rc₂ rc₁ rc₄ rc₃ rc₀ rt₅ rt₂ rt₁ rt₄ rt₃ rt₀
rt₀ rt₀ rt₁ rt₂ rt₃ rt₄ rt₅ rc₀ rc₁ rc₂ rc₃ rc₄ rc₅
rt₁ rt₁ rt₀ rt₃ rt₂ rt₅ rt₄ rc₁ rc₀ rc₃ rc₂ rc₅ rc₄
rt₂ rt₂ rt₅ rt₄ rt₁ rt₀ rt₃ rc₂ rc₅ rc₄ rc₁ rc₀ rc₃
rt₃ rt₃ rt₄ rt₅ rt₀ rt₁ rt₂ rc₃ rc₄ rc₅ rc₀ rc₁ rc₂
rt₄ rt₄ rt₃ rt₀ rt₅ rt₂ rt₁ rc₄ rc₃ rc₀ rc₅ rc₂ rc₁
rt₅ rt₅ rt₂ rt₁ rt₄ rt₃ rt₀ rc₅ rc₂ rc₁ rc₄ rc₃ rc₀

Table four
	second operation
rc₀	rc₁	rc₂	rc₃	rc₄	rc₅	rt₀	rt₁	rt₂	rt₃	rt₄	rt₅
first operation	rc₀	rc₀	rc₁	rc₂	rc₃	rc₄	rc₅	rt₀	rt₁	rt₂	rt₃	rt₄	rt₅
rc₁	rc₁	rc₀	rc₃	rc₂	rc₅	rc₄	rt₁	rt₀	rt₃	rt₂	rt₅	rt₄
rc₂	rc₂	rc₅	rc₄	rc₁	rc₀	rc₃	rt₂	rt₅	rt₄	rt₁	rt₀	rt₃
rc₃	rc₃	rc₄	rc₅	rc₀	rc₁	rc₂	rt₃	rt₄	rt₅	rt₀	rt₁	rt₂
rc₄	rc₄	rc₃	rc₀	rc₅	rc₂	rc₁	rt₄	rt₃	rt₀	rt₅	rt₂	rt₁
rc₅	rc₅	rc₂	rc₁	rc₄	rc₃	rc₀	rt₅	rt₂	rt₁	rt₄	rt₃	rt₀
rt₀	rt₀	rt₁	rt₂	rt₃	rt₄	rt₅	rc₀	rc₁	rc₂	rc₃	rc₄	rc₅
rt₁	rt₁	rt₀	rt₃	rt₂	rt₅	rt₄	rc₁	rc₀	rc₃	rc₂	rc₅	rc₄
rt₂	rt₂	rt₅	rt₄	rt₁	rt₀	rt₃	rc₂	rc₅	rc₄	rc₁	rc₀	rc₃
rt₃	rt₃	rt₄	rt₅	rt₀	rt₁	rt₂	rc₃	rc₄	rc₅	rc₀	rc₁	rc₂
rt₄	rt₄	rt₃	rt₀	rt₅	rt₂	rt₁	rc₄	rc₃	rc₀	rc₅	rc₂	rc₁
rt₅	rt₅	rt₂	rt₁	rt₄	rt₃	rt₀	rc₅	rc₂	rc₁	rc₄	rc₃	rc₀

Based on what can be exchanged with what else in the operations, one can partition the tiles into subsets:

Table five
S₀ 4:0, 4:1, 4:2
S₁ 4:3, 4:4, 4:5
H₀ 6:0, 6:1
H₁ 6:2, 6:3, 6:4
H₂ 6:5, 6:6, 6:7, 6:8, 6:9, 6:A
H₃ 6:B, 6:C, 6:D
H₄ 6:E, 6:F

Table five
S₀	4:0, 4:1, 4:2
S₁	4:3, 4:4, 4:5
H₀	6:0, 6:1
H₁	6:2, 6:3, 6:4
H₂	6:5, 6:6, 6:7, 6:8, 6:9, 6:A
H₃	6:B, 6:C, 6:D
H₄	6:E, 6:F

This will be helpful in the following section.

Each row of table six below shows a set of solutions that is closed under the twelve operations. Most sets have 12 members, but some have fewer.

The members of each set are presented in a particular sequence of operations of its first member. Note that the choice of first member was arbitary; any other member would have been just as good.

Sets of 12 members are written in the order rc₀, rc₁, rc₂, rc₃, rc₄, rc₅; rt₀, rt₁, rt₂, rt₃, rt₄, rt₅.
Certain identities apply to the 4-member sets:
- rc₀ = rc₂ = rc₄
- rc₁ = rc₃ = rc₅
- rt₀ = rt₂ = rt₄
- rt₁ = rt₃ = rt₅
Members are written in the order rc₀, rc₁; rt₀, rt₁.
Further identities arise with the 2-member sets:
- rc₀ = rc₂ = rc₄ = rt₁ = rt₃ = rt₅
- rc₁ = rc₃ = rc₅ = rt₀ = rt₂ = rt₄
Members are written in the order rc₀, rc₁.

Each number refers to the solution's position in the lengthy table.

Taking the first row as an example, 6 = rc₀(6), 63 = rc₁(6), 35 = rc₂(6), 33 = rc₃(6) et cetera. All the tiles for these solutions come from partitions H₀ (6:0 and 6:1), H₁ (6:2 through 6:4) and H₂ (6:5 through 6:A). Tiles 6:B through 6:F are not needed.

Table six
Solutions Partitions
6, 63, 35, 33, 8, 38; 64, 5, 34, 36, 37, 7 H₀, H₁, H₂
1, 90; 91, 3
2, 92; 89, 4
215, 219; 218, 216 H₀, H₂, H₃
213, 217
214, 220
93, 164, 161, 126, 118, 102; 99, 153, 166, 120, 124, 97 H₀, H₁, H₂, H₃
239, 428, 299, 393, 251, 372; 366, 298, 435, 252, 383, 240
226, 423, 309, 394, 263, 367; 365, 297, 438, 269, 388, 233
135, 205, 186, 151, 198, 182; 140, 195, 189, 146, 209, 179
279, 459, 323, 412, 332, 447; 407, 331, 452, 283, 460, 319
228, 424, 312, 397, 260, 370; 363, 293, 439, 273, 385, 234
138, 211, 174, 142, 201, 188; 150, 204, 181, 137, 207, 175
136, 208, 176, 149, 203, 180; 141, 202, 187, 139, 212, 173
225, 425, 311, 395, 261, 371; 364, 296, 437, 270, 387, 236
229, 427, 310, 396, 262, 368; 361, 294, 436, 272, 386, 235
245, 421, 316, 391, 264, 358; 354, 304, 432, 274, 381, 248
237, 422, 315, 392, 277, 357; 353, 314, 431, 278, 382, 238
227, 426, 313, 398, 259, 369; 362, 295, 440, 271, 384, 232
340, 418, 287, 458, 327, 469; 470, 288, 417, 328, 457, 339
94, 163, 160, 125, 117, 101; 100, 154, 165, 121, 123, 96
29, 41, 55, 70, 86, 15; 13, 56, 40, 83, 75, 26 H₀, H₁, H₂, H₄
57, 9, 84, 46, 25, 76; 44, 30, 74, 54, 19, 88
43, 23, 71, 58, 12, 77; 52, 18, 82, 50, 32, 73
11, 60, 48, 81, 67, 22; 31, 47, 53, 69, 78, 16
224, 419, 308, 389, 258, 355; 351, 291, 429, 268, 380, 230
45, 28, 68, 59, 20, 79; 61, 10, 85, 42, 27, 66
51, 17, 87, 39, 24, 72; 49, 21, 65, 62, 14, 80
223, 420, 307, 390, 257, 356; 352, 292, 430, 267, 379, 231
134, 193, 185, 144, 197, 177; 133, 194, 184, 145, 196, 178
221, 350; 349, 222 H₀, H₁, H₃, H₄
344, 473, 346, 476, 347, 471; 472, 345, 474, 348, 475, 343 H₀, H₂, H₃, H₄
244, 443, 300, 403, 253, 374; 373, 301, 442, 255, 404, 241 H₀, H₁, H₂, H₃, H₄
243, 441, 302, 401, 254, 375; 376, 303, 444, 256, 402, 242
286, 465, 322, 410, 333, 453; 413, 336, 450, 281, 463, 324
116, 127, 106, 109, 158, 169; 128, 115, 110, 105, 168, 159
276, 405, 250, 377, 306, 446; 406, 266, 378, 247, 445, 318
148, 210, 191, 143, 199, 190; 152, 200, 183, 147, 206, 192
131, 122, 107, 98, 172, 155; 119, 132, 95, 108, 162, 167
249, 434, 305, 399, 275, 359; 360, 317, 433, 265, 400, 246
280, 461, 326, 415, 338, 448; 408, 334, 455, 285, 466, 320
282, 462, 325, 414, 337, 449; 409, 335, 454, 284, 467, 321
156, 111, 113, 170, 103, 129; 171, 104, 130, 157, 112, 114
290, 468, 329, 411, 341, 456; 416, 342, 451, 289, 464, 330

Table six
Solutions	Partitions
6, 63, 35, 33, 8, 38; 64, 5, 34, 36, 37, 7	H₀, H₁, H₂
1, 90; 91, 3
2, 92; 89, 4
215, 219; 218, 216	H₀, H₂, H₃
213, 217
214, 220
93, 164, 161, 126, 118, 102; 99, 153, 166, 120, 124, 97	H₀, H₁, H₂, H₃
239, 428, 299, 393, 251, 372; 366, 298, 435, 252, 383, 240
226, 423, 309, 394, 263, 367; 365, 297, 438, 269, 388, 233
135, 205, 186, 151, 198, 182; 140, 195, 189, 146, 209, 179
279, 459, 323, 412, 332, 447; 407, 331, 452, 283, 460, 319
228, 424, 312, 397, 260, 370; 363, 293, 439, 273, 385, 234
138, 211, 174, 142, 201, 188; 150, 204, 181, 137, 207, 175
136, 208, 176, 149, 203, 180; 141, 202, 187, 139, 212, 173
225, 425, 311, 395, 261, 371; 364, 296, 437, 270, 387, 236
229, 427, 310, 396, 262, 368; 361, 294, 436, 272, 386, 235
245, 421, 316, 391, 264, 358; 354, 304, 432, 274, 381, 248
237, 422, 315, 392, 277, 357; 353, 314, 431, 278, 382, 238
227, 426, 313, 398, 259, 369; 362, 295, 440, 271, 384, 232
340, 418, 287, 458, 327, 469; 470, 288, 417, 328, 457, 339
94, 163, 160, 125, 117, 101; 100, 154, 165, 121, 123, 96
29, 41, 55, 70, 86, 15; 13, 56, 40, 83, 75, 26	H₀, H₁, H₂, H₄
57, 9, 84, 46, 25, 76; 44, 30, 74, 54, 19, 88
43, 23, 71, 58, 12, 77; 52, 18, 82, 50, 32, 73
11, 60, 48, 81, 67, 22; 31, 47, 53, 69, 78, 16
224, 419, 308, 389, 258, 355; 351, 291, 429, 268, 380, 230
45, 28, 68, 59, 20, 79; 61, 10, 85, 42, 27, 66
51, 17, 87, 39, 24, 72; 49, 21, 65, 62, 14, 80
223, 420, 307, 390, 257, 356; 352, 292, 430, 267, 379, 231
134, 193, 185, 144, 197, 177; 133, 194, 184, 145, 196, 178
221, 350; 349, 222	H₀, H₁, H₃, H₄
344, 473, 346, 476, 347, 471; 472, 345, 474, 348, 475, 343	H₀, H₂, H₃, H₄
244, 443, 300, 403, 253, 374; 373, 301, 442, 255, 404, 241	H₀, H₁, H₂, H₃, H₄
243, 441, 302, 401, 254, 375; 376, 303, 444, 256, 402, 242
286, 465, 322, 410, 333, 453; 413, 336, 450, 281, 463, 324
116, 127, 106, 109, 158, 169; 128, 115, 110, 105, 168, 159
276, 405, 250, 377, 306, 446; 406, 266, 378, 247, 445, 318
148, 210, 191, 143, 199, 190; 152, 200, 183, 147, 206, 192
131, 122, 107, 98, 172, 155; 119, 132, 95, 108, 162, 167
249, 434, 305, 399, 275, 359; 360, 317, 433, 265, 400, 246
280, 461, 326, 415, 338, 448; 408, 334, 455, 285, 466, 320
282, 462, 325, 414, 337, 449; 409, 335, 454, 284, 467, 321
156, 111, 113, 170, 103, 129; 171, 104, 130, 157, 112, 114
290, 468, 329, 411, 341, 456; 416, 342, 451, 289, 464, 330

Every solution uses at least one member of H₀.

A different puzzle can be created by using the fourteen tiles in table seven. It retains five tiles from table one, but introduces nine other tiles with four-stripe arcs.

Table seven

4:0
4:1
4:2
new
new
new

6:0
6:1
new
new

new
new
new
new

Table seven
4:0	4:1	4:2	new	new	new
6:0	6:1	new	new
new	new	new	new

The rule is the same: form a truncated octahedron, and wherever two tiles adjoin match the number of stripes. Here is one solution:

— Figure six —

There are four solutions in total, one obtained from another by applying either or both of these operations:

Table eight
without reflection:

single → single
double → double
triple → quadruple
quadruple → triple with reflection:

single ⇒ single
double ⇒ double
triple ⇒ triple
quadruple ⇒ quadruple

Table eight
without reflection: single → single double → double triple → quadruple quadruple → triple	with reflection: single ⇒ single double ⇒ double triple ⇒ triple quadruple ⇒ quadruple

A property of the truncated octahedron is that, at each vertex, two hexagons and one square meet. What happens if this is changed to one hexagon and two squares? A much simpler puzzle, in the shape of a hexagonal prism, ensues. Here is one solution:

— Figure seven —

All other solutions are related to this one by the rc_n and rt_n operations.

A further step is to have no hexagons and three squares at each vertex, forming a cube. In this case, however, no solution exists. On the other hand, with three hexagons and no squares at each vertex, the shape becomes a plane tessellation and results in the most popular version of the Tantrix puzzle.

Besides squares and hexagons, a polygon that has an even number of sides is the octagon. Among possible ROCK variant shapes are:

the octagonal prism, a polyhedron with one octagon and two squares at each vertex
the truncated cuboctahedron, a polyhedron with one octagon, one hexagon, and one square at each vertex
the truncated square tiling , a plane tessellation with two octagons and one square at each vertex.

With eight sides, an octagonal tile would have four paths, presumably marked with four colors or four multiplicities of stripes. Recall that table five partitions the squares into two subsets and the hexagons into five; for octagons there would be fully eighteen comparable subsets. In table nine are shown one member of each subset and the subset's name.

Table nine
6
permutations
each
O₀
O₁
O₂
12
permutations
each
O₃
O₄
O₅
O₆
O₇
24
permutations
each
O₈
O₉
O₁₀
O₁₁
O₁₂

O₁₃
O₁₄
O₁₅
O₁₆
O₁₇

Table nine
6 permutations each	O₀	O₁	O₂
12 permutations each	O₃	O₄	O₅	O₆	O₇
24 permutations each	O₈	O₉	O₁₀	O₁₁	O₁₂
O₁₃	O₁₄	O₁₅	O₁₆	O₁₇

Because, for most configurations there are 24 permutations of path markings (hence 24 possible tiles), octagonal pieces could easily become too numerous and complicated for enjoyable play. Likely then is that any puzzle using octagonal tiles would include only a judiciously chosen few. Noteworthy is that each of sets O₀, O₁, and O₂ has only six permutations, which is precisely the number of octagonal faces in a truncated cuboctahedron; this fact is perhaps an avenue to an elegant selection of tiles.

By way of example, these are the six members of O₁:

Table ten

Rhombi are used in another variation.

Miscellaneous.

Images were described in Abobe PostScript; two-dimensional images came from hand-written files, while code for three-dimensional images was generated by a C++ program under XCode. The PostScript code was then converted to pdf using pstopdf. The pdf was then converted to gif with Preview. Most of the html was hand-written, although the lengthy table was generated by the same C++ program that calculated the solutions.

Throughout geometry, there are many octahedra, and many ways to truncate them. However, the term truncated octahedron is almost always taken to mean that all vertices of a regular octahedron have been truncated in such fashion that an equilateral tetradecahedron results. Often overlooked is that the edges of a cube can be truncated to give the same figure.

A candidate solution is any arrangement where 8 hexagonal and 6 square tiles have been physically fitted onto the ROCK. whether or not the stripes match at adjacencies. Fans of big numbers will enjoy learning that there are 7,474,701,359,539,814,400 (7 quintillion or 7 trillion, depending on scale) candidates. However, not all candidate solutions need be examined directly: when two adjacent tiles do not match in an incomplete candidate solution, there is no use trying to complete the candidate. The author used a brute-force search that required about 30 hours of programming time and under 5 minutes of running time on an iMac computer.