ICOSIAN

The “Icosian” game was invented in the 19th century by Sir John Hamilton, a famous Irish mathematician. Note that the board is the flattening of the edges and vertices of a regular dodecahedron (a polyhedron with twelve faces, all pentagonal).

On this board, we can propose several challenges, ranging from easy to impossible!

One of the first challenges is finding a path passing through each point exactly once. This type of path in graphs became known as a Hamiltonian path. The path is called a circuit if it returns to the starting point. On Eulerian paths (see Königsberg Bridges), one passes exactly once on each edge.

Start by finding a path that passes only once at each point. Then block one or more points (ie exclude them from the game) and try to do the same.
Image: Alisson Ricardo.
Could it be that there is a pair of points that, when eliminated, prevent creating a circuit by passing each point exactly once?

USEFUL LINKS

This Wikipedia page briefly explains how the Icosian game works and the motivation for its creation, which has to do with the symmetry problems of an icosahedron. The page contains a super interesting image, which three-dimensions the problem!


After trying to imagine and sketch out how to solve this problem, you can watch this maxprimenumberarp channel video, which shows - slowly and step by step - a solution to the initial problem, that is, without placing pieces that interrupt the passage.                                                                                                                

This game largely relates to graph theory, so this UNESC (Extreme South of Santa Catarina's University) article, written by an academic and two university professors, talks about the use of graph theory in the icosian game. A bonus: some of the images in the article were taken from Matemateca: the old board's and the display's!

This Wikipedia page talks about William Rowan Hamilton, the creator of the icosian game. This article brings a dense biography, mentioning several complicated aspects of his life, and describes his achievements in various areas, such as mathematics, physics and astronomy, placing the Icosian game as an original work.

Still talking about William Hamilton and his complicated life, this OpenMind BBVA article talks about an episode of "mathematical vandalism" linked to Irish. In addition, it brings some challenges related to the Icosian game and similar structures. Try it and see if you can do it!

Finally, this Wikipedia page talks about hamiltonian paths, explaining what they are and mentioning some properties. In addition, it gives some examples of where you can see them, as in some types of graphs and in our icosian game - as the images on the page illustrate!