Tantrix

Introduction
Original Tantrix Puzzle (Crazy Tantrix)
About the background of this page
Tantrix Xtreme Puzzle
The Super 5 Puzzles
Three Pyramid Puzzles
My own puzzles
The Rock
Unsolved(?) puzzles

Hints on solving a loop puzzle
Solution to Original Puzzle
Solution to Tantrix Xtreme
Solutions to Super 5 puzzles
Solutions to the Pyramid Puzzles
Solutions to my own puzzles
Solutions to the Rock

Introduction
Tantrix is the brand name for a set of hexagonal bakelite tiles. Each tile has three lines, of different colours, which go from one side to another. Some sets of tiles can be use as puzzles, in which you have to arrange the tiles so that the edges match in colour, thus forming coloured lines or even loops. The tiles can also be used for playing a game with two or more people. On this page however, I will only discuss the puzzles.

Tantrix uses only four shapes of tile:

:
Three sharp bends or 'corners'	A straight and two corners	A straight and two shallow bends	A corner and two bends

Tantrix seems to be based on an older tileset, created some 40 years earlier by Charles Titus and Craige Schenstedt. It was marketed then as Psychepaths, but is now made and sold by Kadon Enterprises, Inc. with the name Kaliko. The Kaliko tiles form a complete set of all possible path patterns with 3 colours. It differs from Tantrix only in that a colour is may be repeated on a tile, and the tile type with 3 straights is used as well.

Original Tantrix Puzzle (Crazy Tantrix)
The first version of Tantrix was a puzzle consisting of 10 hexagonal tiles. The aim is to tile the pieces (without leaving holes) to make a single loop using all the parts of one colour and such that the edges of all adjacent tiles match colours. Any colour loop is possible.

If we use the letters R, B, and Y for the colours (red/blue/yellow), then a tile can be described by a list of six such letters which denote the colour at each of the edges clockwise around the tile.

These are the tiles:

1. Three sharp bends or 'corners':
BBRRYY (30|14)
2. A straight and two corners:
BBRYYR (32|5)	BBYRRY (31|13)
3. A straight and two shallow bends:
BRYBYR (36|4)	RBYRYB (35|9)	YBRYRB (34|6)
4. A corner and two bends:
BBRYRY (41|8)	BBYRYR (42|7)	YYBRBR (37|1)	YYRBRB (38|10)

Note that four more tiles are possible with 3 colours but not included in the set: YYRRBB (3 corners but mirror image of the above), BRRBYY (straight blue and two corners), RRBYBY and RRYBYB (red corner and two bends).

Therefore there are only 14 possible tiles with these three colours, because the type of tile with three straight (e.g. YRBYRB) is not used. With an extra colour (green) there are 4·14=56 possible tiles. Such a full set is available, and allows for many other puzzles as well as a clever multi-player strategy game. The numbers in brackets are the numbers of the tiles using the standard 1-56 numberings that Tantrix uses.

Note that the original Tantrix puzzle may have a different colour combination than that above. For a present I received second set of 10 Tantrix pieces, but these used green instead of yellow. Other than that they are exactly the same. It is possible to combine the two sets, to make a larger puzzle. Below for example is a solution for a large red loop. A blue loop is more difficult, and I leave that to you.

Solutions to the original puzzle
Hints on solving loop puzzles

About the background of this page
The background image used on this page consists of a repeated pattern using the ten tiles of the Original Puzzle. It was quite difficult to find a ten-tile shape that can be tiled easily. I do not believe that there is such a shape that tiles the plane with only translations, and that only shapes that need rotations as well are possible. What makes it harder is that the background image must be rectangular. In the end I plumped for the triangle shape shown on the right that combined with an upside down copy of itself forms a 4 by 5 diamond. By making sure its edges were such that the diamond could be tiled in a regular rectangular manner, the eventual background image would remain fairly small. Click the image to see the rectangular repeated pattern. I found several solutions, but this one is the only one I found that has infinitely long lines of all three colours, though it does have a loop. I did not find any with only loops or only lines that use ten tiles, but have not done an exhaustive search.
I did find a very nice 9-tile shape (3x3 diamond) that tiles the plane (without rotations) and has only loops, and another that has only lines. The trouble with these however is that they do not tile in a rectangular way, so the background image would have to be quite large and contain many copies if this shape (18 in fact), or be skewed in some way. The tiling patterns are shown below. Click on them to see the large rectangular tile that would be needed for a background.

Tantrix Xtreme Puzzle
Tantrix Xtreme is a puzzle version of Tantrix similar to the original, but more challenging. It also consists of 10 hexagonal Tantrix tiles, but now four colours are used in the set, and the tiles are numbered on the back. The tiles are shown here:

1. BBRGRG (55|38)	2. BBRGGR (46|30)	3. BBGGRR (43|25)	4. BGRGBR (49|42)	5. BBGRRG (45|26)
6. RGBGRB (50|40)	7. RRBGBG (53|27)	8. RRYBYB (40|12)	9. BRYRBY (34|6)	10. BBYRYR (42|7)

Note that the exact colours may vary. In my set it is more purple than red, and uses white instead of yellow. For consistency I have used the four standard colours on this page however.

There are all together 10 challenges, from easy to rather hard. Start with the tiles numbered 1-3, and make a single loop of one colour. The same rules apply as with almost all other Tantrix puzzles - all adjacent tile edges should match colour, and there should not be holes in the tile arrangement. Once you have done a loop with tiles 1-3, then try to make a loop with 1-4, and then 1-5 and so on until you eventually use all ten tiles to build a loop.
In most cases only one of the colours can be used for making the loop, the others being impossible. In some cases, in particular the 10 tile loop, more than one colour loop is possible.

After those eight loop challenges, there are two difficult line puzzles. The aim is to put the ten tiles in a triangle shape, but with the red or the blue colour forming one long line through all the tiles. Both red and blue have solutions.

Solutions to the Xtreme Tantrix
Hints on solving loop puzzles

The Super 5 Puzzles
Some years ago I acquired the full set of 56 tiles. At that time the numbering of the tiles was given in the accompanying booklet. Nowadays a different numbering is used, and the numbers are engraved on the backs of the tiles. Below are conversion tables for the two numberings, also listing all the tile patterns.

Old     New     Colour           Old     New     Colour
37       1      YYBRBR            1      21      GGRRYY
33       2      RRBYYB            2      23      GGYYRR
29       3      BBYYRR            3      16      GGYRRY
36       4      BRYBYR            4      22      RRGYYG
32       5      BBRYYR            5      15      GGRYYR
34       6      YBRYRB            6      35      YGRYRG
42       7      BBYRYR            7      32      GRYGYR
41       8      BBRYRY            8      34      RGYRYG
35       9      RBYRYB            9      18      YYRGRG
38      10      YYRBRB           10      17      YYGRGR
39      11      RRBYBY           11      31      GGRYRY
40      12      RRYBYB           12      33      GGYRYR
31      13      BBYRRY           13      19      RRGYGY
30      14      BBRRYY           14      20      RRYGYG

 5      15      GGRYYR           15      43      BBYYGG
 3      16      GGYRRY           16      45      BBGGYY
10      17      YYGRGR           17      47      BBYGGY
 9      18      YYRGRG           18      49      BBGYYG
13      19      RRGYGY           19      48      GGBYYB
14      20      RRYGYG           20      44      YBGYGB
 1      21      GGRRYY           21      51      GBYGYB
 4      22      RRGYYG           22      50      BGYBYG
 2      23      GGYYRR           23      54      YYBGBG
                                 24      53      YYGBGB
47      24      GGBRRB           25      46      GGBYBY
43      25      BBGGRR           26      52      GGYBYB
45      26      BBGRRG           27      55      BBGYGY
53      27      RRBGBG           28      56      BBYGYG
44      28      BBRRGG
54      29      RRGBGB           29       3      BBYYRR
46      30      BBRGGR           30      14      BBRRYY
                                 31      13      BBYRRY
11      31      GGRYRY           32       5      BBRYYR
 7      32      GRYGYR           33       2      RRBYYB
12      33      GGYRYR           34       6      YBRYRB
 8      34      RGYRYG           35       9      RBYRYB
 6      35      YGRYRG           36       4      BRYBYR
                                 37       1      YYBRBR
52      36      GGRBRB           38      10      YYRBRB
56      37      BBGRGR           39      11      RRBYBY
55      38      BBRGRG           40      12      RRYBYB
48      39      GBRGRB           41       8      BBRYRY
50      40      BGRBRG           42       7      BBYRYR
51      41      GGBRBR
49      42      RBGRGB           43      25      BBGGRR
                                 44      28      BBRRGG
15      43      BBYYGG           45      26      BBGRRG
20      44      YBGYGB           46      30      BBRGGR
16      45      BBGGYY           47      24      GGBRRB
25      46      GGBYBY           48      39      GBRGRB
17      47      BBYGGY           49      42      RBGRGB
19      48      GGBYYB           50      40      BGRBRG
18      49      BBGYYG           51      41      GGBRBR
22      50      BGYBYG           52      36      GGRBRB
21      51      GBYGYB           53      27      RRBGBG
26      52      GGYBYB           54      29      RRGBGB
24      53      YYGBGB           55      38      BBRGRG
23      54      YYBGBG           56      37      BBGRGR
27      55      BBGYGY
28      56      BBYGYG

In the booklet I got with the set, several puzzles were given. The full set can be separated into 5 separate sets, which are called the 'Super 5' puzzles. They are called Junior, Student, Professor, Master, and Genius. These were sold separately as well.

In the Junior, Student and Master puzzles the aim is to make a loop of one colour, just like the Original puzzle. Of course, the loop must include all the tiles with that colour, and the tiles may not enclose a hole. In the Professor puzzle the aim is to make two simultaneous loops. Again each loop must use all the tiles of that colour. Tiles which have both chosen colours must therefore be incorporated in both loops. The Genius puzzle is extremely difficult, as now you must make two lines instead of loops. The restrictions that loops impose do not apply to this puzzle. It has two types of solutions (red/blue, red/yellow), and it took me about a week to find a solution for each. I thought there was essentially only one solution of each type, but Alexander Fronk sent me a second solution for the red/yellow lines.

Solutions to Super 5 puzzles

Three Pyramid Puzzles
Three puzzles that are mentioned in the booklet of the Tantrix game use 15 tiles, and they all have solutions in a pyramid shape, i.e. a size 5 triangle. In puzzle 1 you have to make a red loop, in puzzle 2 two loops using any two colours (like the Professor puzzle), and in puzzle 3 two lines of any two colours (like the Genius puzzle). Puzzle 3 is extremely hard to solve. The tiles to use are listed below:

Solutions to the Pyramid Puzzles

Later versions of the instruction manual may have listed other kinds of puzzles.

My own puzzles

• The Master puzzle above has many solutions. I found about 30 solutions in one evening. The loop in a solution has both convex and concave parts. Usually three tiles are involved in the concave parts (the 'dents' if you like). As far as I know there is only solution in which only two tiles are part of the concave sections of the loop. Find it.
• The unsolved line problem can also be done using only three colours. In other words, using the 14 tiles with only three of the colours make a shape with the longest three lines. Only the longest line of each colour counts, and the aim is to make the sum of the three lengths as large as possible. The highest possible total length is 32, and I have two solutions to it.
• From the full set, take the 12 pieces which have a straight line and two bends. Try to make the shape shown on the right - a triangle with sides of length 5 but with its tips missing. In other words, the shape with rows of lengths 2,3,4,3.
• Do the same as the previous puzzle, but with the 12 pieces which have a straight line and two corners.
  
• Take the 8 pieces which have 3 corners, and any single piece with a corner and two bends (e.g. RRBYBY). Try to make any shape containing 5 circles.

Solutions to my own puzzles

The Rock:
The Rock is a three-dimensional version of Tantrix. It has the shape of a truncated octahedron, which has 8 hexagonal faces and 6 square faces. There are hexagonal and square Tantrix tiles which attach to these faces, and the aim is of course to place all these tiles so that all the coloured lines match up. This means that each colour will be one or more loops on the surface of the rock. Instead of taking the tiles off completely and trying to solve it, it is also possible just to rotate the tiles in place which makes for an easier puzzle.
The tiles are as follows:
Hexagonal tiles:
BBYRYR, BBRYRY, YYBRBR; BRYBYR, RBYRYB; RBBRYY, BRRBYY; BBRRYY
Square tiles:
BRBR, BYBY. RYRY, BBRR, BBYY, RRYY
This puzzle has only 5 solutions.

Solutions to the Rock

Unsolved(?) Puzzles:
There are two so-called 'unsolved' puzzles, using all 56 tiles.

1. The first is to find an arrangement with the four longest lines, one of each colour. Only the longest line of each colour counts, and the best is the one with the longest total length. The current record according to the Tantrix homepage is 146=40+37+35+34. Note that since all possible tiles are in the set, any solution can be rearranged such that the colours are swapped around.

It can be proved that 146 is the maximum attainable, so this puzzle has actually been solved.
Sketch of proof: Consider an arrangement with 4 lines of total length 146. A line of length n uses n tiles, and therefore involves 2n tile sides. Of these, 2n-2 sides are internal, and 2 sides are the endpoints. The 4 lines will therefore use 2·146-8=284 internal sides of tiles. The 56 tiles have 6·56=336 sides all together, leaving 336-284=52 sides along the outside of the arrangement. Longer lines would leave fewer external sides. The size 5 regular hexagon with one edge shaved (i.e. a hexagon with sides 5, 5, 5, 4, 6 and 4) is the shape with the smallest possible perimeter, namely 52. Therefore the lines cannot be longer than 146.
The difficult step is proving that the hexagon shape has the smallest perimeter. I have proved this, in a long and tedious manner as follows: first show that the best shape is nearly convex, i.e. has at most one tile with one external edge. Then write the number of tiles in each row in a list, and show that with the best shape the list will look something like this: 7, 8, 9, 10, 9, 8, 5, i.e. first strictly increasing then decreasing and with successive differences equal to 1 except possibly for the last one. This leaves only a relatively small number of possibilities, and of these the sequence 6, 7, 8, 9, 10, 9, 8, 7, 6, 5 is the best.

2. The second unsolved puzzle is to find the arrangement with the four longest loops. The current record is 136=38+35+33+30.

It can probably be proved that 136 is the maximum attainable.
Sketch of proof:Any shape with 56 tiles must have some tiles with an odd number of external sides, because they cannot be arranged in a triangle (tiles with 4 external edges are at a 60 degree corner, with 2 external edges along a straight side of the arrangement). Each tile with an odd number of external sides must have at least one line connecting to an internal with an external side. Thus any such odd corner will be the start of a line leading into the arrangement, wasting internal sides which are better used for forming loops. The best shape is therefore one with a small perimeter, but which has few odd corners and has its odd corners close together.
Consider a size 10 triangle with one extra tile added anywhere on its side. This has perimeter 62, and its two odd corners are adjacent so these waste only 2 internal sides. This leaves 336-64=272 internal sides for a total loop length of 272/2=136. Any other shape with more odd corners that is more convex will have a smaller perimeter but all internal sides gained are probably all wasted on the lines between the odd corners. It is likely that no other shapes have more available internal sides.
Again the difficulty is proving that this shape is the best. I have not properly proved this.

Hints on finding a solution to any loop puzzle:
Before you read on, please be aware that these hints will make it quite easy and may spoil your enjoyment of the puzzle.

1. First decide which colour you are going to make the loop from. If you are doing the original puzzle, any colour can be solved. The Super 5 puzzles Junior/Student/Master (see below) only allow one colour to be looped. It is not hard to decide which colour to use. For example, if a colour has an odd number of bends it cannot be looped. Similarly if two colours cross an odd number of times, neither colour can be looped.
2. A good strategy is to make a loop of your chosen colour, completely ignoring the other colours. Then swap pairs of pieces that leave the loop intact but fix the other lines so that the colours match up.
3. In my first original puzzle, yellow crosses with blue on only two tiles. Therefore when making a blue loop, a yellow line must enter and leave the loop using the two crossing tiles available. Similarly, a yellow loop must have a blue line entering and leaving using those two tiles.
4. Every tile which lies mostly inside a blue/yellow loop will also have to have the yellow/blue line going over the tile.
5. When making a blue/yellow loop, use hints 3 and 4 above to plot how you want the yellow/blue line to run. Note that the crossing over the straight edge is fixed because there is only on straight part, but the crossing over the bend can be placed elsewhere on the loop. Use swaps to place the line along the path you plotted. The rest of the tiles inside the loop will then match up.
6. Once the loop and its insides are correct, make swaps of tiles which lie mostly outside the loop to correct any faults until all tiles match up.
7. If you make a red loop, there are four yellow and four blue crossings. In this case choose one of the two colours, and plot two lines that go into and out of the loop, and that together go through every tile that lies mostly inside the loop.

Solutions to Original Puzzle
Here are three solutions of the original Tantrix puzzle, one of each colour. There are dozens of possibilities for each, so these are only examples.

Here is a solution for a large red loop with two sets combined. A blue loop is a little more difficult, and I leave that to you.

Solutions to Tantrix Xtreme
Here are solutions to the 8 loop puzzles. Two solutions are given for the 7 and the 10 tile loop, because two colours are possible. In all the other cases, only one of the colours can be used to make a loop.

1-3, blue	1-4, green	1-5, green	1-6, red	1-7, green
1-7, red	1-8, red	1-9, blue	1-10, red	1-10, blue By Michael Gegenwart

Here are the solutions to the 2 pyramid puzzles. There are essentially three solutions for a red line and only one for a blue line. These are shown below.

Minor variations are possible by rearranging one or more sets of tiles. For example in the first solution you could swap tiles 2 and 4, or swap tiles 5 and 7, or turn tile 1 around a bit, which leads to 8 variations. In solution 2 only tiles 3 and 8 on the corners can be trivially swapped. In solution 3 we can swap 3 and 5 as well as the trivial corners 2 and 6. There are six variations of the blue solution by rearranging the tiles from left and right columns (tiles 1-5).

Solutions to the Super 5 Puzzles:
While the professor and Genius solutions are essentially unique, the others have other solutions than those given here.

Junior loop puzzle. Only a blue loop is possible.

Student loop puzzle. Only a green loop is possible.

Master loop puzzle. Only a green loop is possible.

Professor double loop puzzle. Only blue/yellow loops are possible.

Genius double line puzzle. The red/yellow solution 1:

Genius double line puzzle. The red/yellow solution 2: By Alexander Fronk.

Genius double line puzzle. The red/blue solution:

Solutions to the pyramid puzzles:

Pyramid Puzzle 1. Red loop.

Pyramid Puzzle 2. Red/Yellow loops.

Pyramid Puzzle 3. Red/Blue lines.

Solutions to my own puzzles:

Master Puzzle Loop with only two concave pieces	Three line puzzle Total length= 9+11+12=32
Straight + Bends	Straight + Corners	Five Rings

Solutions to the Rock: