The group discussion on the methods of solving the different challenges using Involution© can be complemented by an evaluation. It contributes to the consolidation of the knowledge and skills acquired.

Create a solution book for Involution© in 3 colours

Let’s play Involution© with 3 colours: 2 black rings, 2 white rings and 1 hole without a ring (the hole without the ring plays the role of the third colour). The pupils are responsible for writing a booklet containing optimal solutions for a number of challenges (determined by the teacher). The choice of presentation is up to the pupils – they can be simple descriptions of the processes, drawings, purely visual representations or a video produced by the pupils themselves.

To find the optimal solutions, the pupils must build the graph corresponding to Involution© with 3 colours and look for the shortest path in this graph (careful – this path may not always exist!).

The completed solutions booklet can then be exchanged among the pupils and checked for accuracy against the game. The pupils can then assess each other and give feedback.

It is also possible to do larger and more complex projects. Here are three different ones: the first is a more computer-science based one (and requires programming), the second one falls more into the area of social sciences and creativity and requires writing skills and the third is intended for pupils who are more interested in mathematics.

Programming Involution©

The goal is to write a programme (on Scratch or Python), using the algorithm established in challenges 15 and 16 to solve the Involution© game.

There are several levels of difficulty:

Write a programme that takes as the input two configurations of the Involution© game and gives as the output the minimum number of movements to go from one configuration to another.

Write a programme that takes as the input one configuration of the Involution© game and gives as the output a sequence of moves (not necessarily the sequence corresponding to the optimal solution), allowing to go from this configuration to the configuration where all the white rings are on one side and all the black rings are on the other.

Write a programme that takes as the input one configuration of the Involution© game and gives as the output the optimal sequence of moves allowing to go from this configuration to the configuration where all the white rings are on one side and all the black rings are on the other.

Invent a game for two players

The pupils should think about of a two-player version of Involution©. They must invent a collaborative version and a competitive version of the game. They are then invited to write clear rules of play. The choice of presentation is up to the pupils (small booklet, drawings, digital media, etc.).

And with 3 colours?

Let’s play Involution© with 3 colours: n black rings, n white rings and 1 hole without a ring (the hole without a ring plays the role of the third colour). Here n can take any value greater than or equal to 2. The question the pupils have to answer is the following: from which value of n on can we obtain all the configurations starting from any other configuration?