Batch 2 - Class 307 - Topology: Tic-Tac-Toe on Mobius, Torus
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Preclass Exercise:
- (in class today)
Attendance: Anika, Vansh, Rhea Chadha, Ryan Chadha, Adyant, Aarushi, Ekagra, Siddharth, Ayush, Dhriti, Advay, Tarush, Kabir, Siddhant, Vivaan, Kushagra, Nikhil, Raghav, Shikhar, Yatharth, Aneesh
Class Notes:
Cutting Cylinders and Mobius strips
- Recap some of the cutting exercises that were done
- Cylinder through the middle - two cylinders
- Half-twist mobius through the middle - four-twist loop
- Half-twist mobius through one-third width - four twist loop entangled with half-twist mobius
- Double-twist mobius down the middle
- Three-twist mobius down the middle
- Two cylinders at right angles, down the middle
- Two half-twist mobius at right angles, down the middle - same orientation versus opposing
- Illustrate the above using a mathematical model (a strip on a surface with arrows at the ends to determine the crossover behavior).
- Use this model to predict what will happen if you paste a cylinder with mobius strip and cut that.
- Try to cut this combination and see what actually happens
- Or paste three mobius strips with one cylinder
Tic-Tac-Toe
- Get students to play tic-tac-toe to illustrate that it is a trivial game, with a guaranteed draw
- How would you play the tic-tac-toe game on a cylinder - using the mathematical model we used above. Lets students play a few games. Attempt the following puzzle, and win the first game in one move (X starts)
- How about playing tic-tac-toe on a mobius strip? We can create a mathematical model such as below to think about it.
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- Play to win the following games. First game can be won in one turn.
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- How about playing the game on a torus - a torus is nothing but out cylinder joined on edges. Show students torus, and the construct a mathematical model with them.
- Can students find the similarly of this with any of the previous problems? Give them the same problems to see if they can solve it on torus, and draw a parallel with the cylinder solution. Can they prove that the torus and cylinder problems are the same?
- Lets solve the following puzzle on a torus, but now we are trying to find a line of 3 on a 4x4 board
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- Are any starting positions better than others on a torus? Does the game have a guaranteed finish, or winning strategy?
Homework
- Explore the Klein bottle and projective plane puzzles on https://chalkdustmagazine.com/features/topological-tic-tac-toe/
References: