Trippples: Strategy and Combinatorial Mathematics
Trippples (with three “p”s, p-p-pick up a p-p-penguin fans) is a chess-like strategic game from the 1970s. I found a copy in a Sue Ryder charity shop for £4. Bargain!
Two players, or two teams of two, try to get their transparent marker from one corner of the board to the other. Every square on the board has arrows pointing in three directions. These arrows tell you which directions the pieces can move each turn. The genius touch of Trippples is that your available moves are determined by the arrows under your opponent’s (or opponents’) marker(s). To win, you must force your opponent to move to a square that allows you to move to your destination.
The game comes with lots of rule options. There’s the “fog of war” version, where the arrow tiles are randomly placed upside down at the start and are only revealed the first time someone lands on one. There’s the “strategic placement” version, where players take it in turns to place randomly drawn tiles on the board before starting play. Or there’s the “ultimate” version, where you can start moving your piece around the board even before all the tiles are placed. My partner and I obviously tested our mettle on the ultimate version. She’s been kicking my arse…
With only an eight-by-eight grid to work on, you often find yourself going round in circles trying to find an opening that gets you over the finish line. A good memory and being unconcerned by feelings of deja vu are probably pre-requisites to play.
There was even a US patent on the game, granted back in 1974: “A game comprising a playing board, a plurality of movable playing tiles [placed to] form the playing surface of the game, each playing tile having on its visible surface two or more directional indicia indicating limits on the movements of playing pieces and each set of indicia being different from the set of indicia on any other playing tile.” (paraphrased from the original)The US Patent Office found similar ideas for games going back as far as 1894 in US Patent 519326. The name of this game is lost to history, if it was ever produced at all, and requires a giant board of 15×15 squares. Players drive their pieces to the centre to do battle and then run the gauntlet to reach their opponent’s corner. The creator tries hard to make it sound exciting!
One special thing about Trippples, mentioned in the claims of the patent, is that every tile has a different set of three arrows – “directional indicia” in patent speak. This made it a good opportunity to refresh my rusty knowledge of combinations and permutations. I find this sort of thing exciting even if 99% of the world doesn’t!
According to Wikipedia, combinations are where you have “n” things and take “k” at a time without repetition. The number of different combinations is given by n! / (k! * (n-k)!). In Tripples, n=8 since there are 8 directions any arrow can point. k=3, since there are three arrows on every tile. So, the number of combinations is 8!/(3! * 5!) = 40,320 / (6 * 120) = 56. Tripples should have 56 different tiles.
The Tripples board is the same as a chessboard – 8×8 making up 64 squares in total. The 4 corner squares are the start and destination points for each player, so that brings us down to 60 squares. To get to 56 arrow tiles, 4 of those 60 squares are filled with blank tiles. In the game, blank tiles are either neutral zones that players can’t enter, or are bonus spaces that allow a player to make a second move.
To be honest, the game manual is a bit hazy about what to do with the 4 blank tiles. I can’t help feeling the inventor, a researcher of psycho-cybernetics with the amazing name of William T. Powers, was a bit upset that the realities of mathematics messed up the perfection of his game! The inventor was this William “Bill” Powers, who sadly passed away in 2013, so now we can never be sure.