The Myth of Big Tangle 10x10

When the original tangle puzzles were released, there were four sets of them, all based on the same pattern. The only difference between them is, that each time a different piece is doubled. This piece is:
Tangle1:    Tangle2:    Tangle3:    Tangle4: 
Hence, from the solution of Tangle1, it is immediate to derive solutions for the other tangles: just cycle the colors!
There was a rumour (apparently claimed in the booklet accompanying the tanglebox) that with the four puzzles together, it is possible to make a tangle of 10x10.
This was discussed to be impossible in 1992 and 1996 in the Cube_Lovers forum (ftp://ftp.ai.mit.edu/CubeLovers).

In 1999/2000, (unknowing the discussions of the Cube_Lovers), Sieuwert van Otterloo and I restarted some investigations.
As we both only have Tangle1, we first made speculations about what other types of rope could be used on the other three.

After we found out (the disappointingly unimaginative way) how the real Tangles 2-4 are derived, we started writing programs to search the solution of the mysterious big tangle.
As this risked to take much computer time, Sieuwert made a distributed computer research program (12 fellows joined their computers).
All in vain!
My program, on a 300MHz PC, took 15 days to find ... that there is no solution!

Then we exchanged again our opinions speculating how this scandal could arrive:
perhaps Rubik (or the producer, or -equivalently- the person that proved that there is a solution) chose the wrong set of tangles!
Indeed, with certain (different) sets of 4 tangles, there are one or more solutions.

As an example, if we replace the original set 'Tangle2' by a variant ('Tangle5': ), then the set of four tangles 1,3,4,5 has 17 solutions; most of them (but not all) are very symmetrical.
Here is one of them:
Note that
1. this solution is composed of 6 composed tiles of 2x2 pieces, one of these composed tiles is repeated 5 times along the diagonal from top-left to bottom-right; the other composed tiles are repeated on the lines parallel to the diagonal.
2. if the solution would be extended with a sixth repetition of each of the 6 composed tiles, then the board of 12x12 is itself a tile that can be repeated infinitely in both directions.
Patrick Hahn - (7.4.1) - Homepage - [Puzzles] [Fractals] [Anagrams] [Software] [Humor] [Photos]