UK and world records
Angharad Taylor of Canterbury broke the world record for the Tantrix Four Longest Lines puzzle on 20 September 2006 with a solution that scores 144, just 2 less than what we believe to be the theoretical maximum.
The previous world record holder was Éva Deákné Terkovics from Hungary who, in 2005, found a 143-solution. Before that, Rob Morton of Knaresborough, Yorkshire, was the last British player to hold the world record with a score of 142. Angharad, Éva and Rob have all sent pictures of their records which have been thoroughly checked, but we have not reproduced them here because now we are getting so close to the 146 maximum, we don't want to give you too many clues!
The first British world record of 140 by Jack Kuipers of London can be seen above to give you the idea of how the puzzle works. The world record has progressed very slowly since 1997, when the unaided 'human' record was broken by Yoseph Phillips of Israel with a score of 139 which exceeded the previous best score by 11 points. Craig Garvie of Edinburgh got within one of this record in December 2001 but it was not until Jack came up with his 140 solution a year later that Yoseph's 5-year-old record finally fell.
The Four Longest Lines puzzle, one of the two hardest Tantrix puzzles ever invented, was first fully solved with the aid of a computer by Jamie Sneddon and Paul Martinsen of New Zealand in May 1998. However, no one has managed to find an ultimate 146 solution without the use of a computer in the eight years since.
How it works
The Four Longest Lines puzzle uses all 56 tiles and can be made in any shape, though we believe that only one shape is possible if you want to find one of the ultimate solutions scoring 146. You must try to make four long lines simultaneously, one of each colour. Naturally, all touching links have to match in colour (even those that are not part of the four main lines) and your solution must not contain any holes, i.e. empty spaces surrounded by six tiles. Your score is the combined total of the number of tiles in each of the four lines and only the longest line of each colour counts.
To claim a new record if you live in the UK or Ireland, you need to send a photo or scan of the tiles in your solution to Tantrix UK Ltd, 119 Cedar Road, Croydon, CR0 6UJ, or email it to julia@tantrix.co.uk. We reserve the right to reject record claims if we have any reason to be at all suspicious that an attempt is a copy of a computer solution.
Computer-aided solutions
As mentioned above, Jamie Sneddon and Paul Martinsen of New Zealand came up with the first ever computer-aided ultimate solution in May 1998.on 1 September 1998. They commented:
"Once we had decided upon the best shape, a nearly regular hexagon, and what sort of pieces were allowed at the edges and corners, I wrote a program to fit tiles into the big hexagon, searching for an arrangement that did not contain any loops. On my computer (a 133Mhz Pentium) the program took more than 50 hours to find the solution, during which time it added and removed tiles more than 4 billion times!"
Jaap Scherphuis has made the following comments on whether or not 146 really is the highest possible score for this puzzle:
"It can be proved that 146 is the maximum attainable
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 2ˇ146-8=284 internal sides of tiles. The 56 tiles have 6ˇ56=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."
Go to the main Tantrix UK & Ireland records page
