Culica Mathematical Gems

Mathematical Culica Gems

Not all mathematics is done with numbers. A lot of mathematics deals with patterns and how to solve puzzles that might not look at all mathematical to the layman. With Culica the mathematics is with relations of colour and shape.

The Culica is a fun game. On a different level, the Culica is a concrete (OK, plastic) mathematical object! The Culica games rules are abstract mathematical objects. Where the latter acts on the former, you have a mathematical paradise!

Culica games rules have geometric, combinatorics and optimisation aspects to them. And they also harbour even more nuggets of mathematical gold.

• CuWizard (group theory)

• CuMadness (logic)

• CuLoops (logic)

• CuColours (four colours theorem)

• CuRing (logic, group theory)

• CuFrog (graph theory)

• CuMolecula (logic, group theory, topology)

• CuMatch (group theory, set theory, transformations)

• CuCombat (game theory when played with three or four players)

• CuRow (game theory when three or four players)

• CuPoker (probability)

• CuSol (graph theory)

• CuAnt (random walk, graph theory, probability)

• CuNitrogen (logic, group theory topology)

The games are available to download here: Culica Games Rules

I really like all the games for their playability and mathematical concepts. If I had to chose personal favourites, my favourite Culica puzzles are those that have supreme elegance and elements of surprise and beauty. They also have salient mathematical themes.

• CuWizard

• CuMadness

• CuColours

• CuRing

• CuNitrogen

I have solved all of those "favourite" Culica puzzles. In each case, I did not know whether a solution exists, and I found them tricky to solve, sometimes extremely so. So when I did solve them, it was awesomely satisfying because not only did a solution exist, but it was not easy to find. Sometimes I thought, Nature will have a solution, the puzzle is too beautiful for no solution to exist!  Below are some notes and analysis.

CuWizard – this is rather analogous to Soduku, but played with rows of colours instead of squares of numbers, and with six faces of a cube instead of a flat grid. Apart from trivial transformations, I believe that there is only one solution to this puzzle.

CuMadness – This is a logic puzzle, and I struggled with it. I had practically given up on it (not knowing if a solution even existed), and was about to settle for nearly complete, when I spotted an error in my latest attempt. In fixing the error, I solved CuMadness, a euphoric moment. Solving this puzzle (and variations of it) feels rather similar to solving “Logic Problems” puzzles. If this colour is next to that colour and that is next to...

CuLoops – This is also a logic puzzle, and is very profound indeed. It is connected to another game called CuLook - see here for some surprisingly fun mathematics.

CuColours - This is the first great puzzle I invented. Four colours. Cover the cube without colours touching. It's tricky! This puzzle is reminiscent of the notorious Four Colour Theorem, so I thought, at first, that it must therefore be possible to solve CuColours – and I did manage to solve it. Later I realised that, because even diagonally connected pegs of the same colour are not allowed, CuColours is not covered by the Four Colour Theorem. In a sense it is harder! :-)

More recently, I joined two Culica cubes together and solved CuColours on that 6x3x3 cuboid. Having done that, it was trivial to show that any 3n x 3 x 3 cuboid be solved with CuColours rules, where n is an integer > 0. It is also solved with three Culicas joined in an L configuration.

CuRing - This puzzle is a little gem, with only one solution to this puzzle (aside from trivial transformations), I reckon.

CuNitrogen - Another little gem, this has two exact solutions, and only two, I believe (aside from trivial transformations).

CuFrog – CuFrog has a relative, CuKnights, which works the same as CuFrog but with an L shape. CuKnights is a chess knight's tour of the Culica. Both CuFrog and CuKnights have solutions, but I have not solved them personally.

There is lots more to say on the games and puzzles, but I'll keep it brief.

I have not programmed the Culica into a computer to find the solutions to puzzles. That is cheating, but also a nice programming challenge in itself!

The Culica makes a great mathematics teaching aid for students of all ages. Also some of the games may be a source of new mathematical research. Mathematicians will notice that a lot of these games are specific instances of rules that can be generalised in many ways.

Some of these generalisations will inspire new Culica game variations and puzzles. I have already invented quite a few and these will be published on this website in the not-to-distant future.

Special mention: this article on the fun mathematics of two Culica puzzles - the mathematics of CuLook and CuLoops.

 

If you're a mathematician, a teacher. or anyone else, and have ideas or would like to be involved in the Culica project in some way, do get in touch!