Batch 2 - Class 291 - Game of coins
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Preclass Exercise:
- MartinShCol - 1.14 - There are bowling pins of red color and black color. Can you select 10 pins from these and place them in position of bowling pins, so that no equilateral triangle has all vertices of the same color?
- Answer: This is not possible. Progressively placing the pins starting from the center position helps arrive at the conclusion.
Attendance: Raghav, Kabir, Ayush, Rhea Chadha, Ryan Chadha, Rohan, Yatharth, Anika, Ekagra, Mihir, Advay, Siddharth, Vivaan, Nikhil Pande, Aarushi, Aarav, Aarkin, Adyant, Aneesh, Shikhar, Kushagra, Vansh, Aarush
Class Notes:
Review Game of Hex
In Hex, we claimed that the first person has the winning strategy. How did we do it? By "Strategy stealing" argument. Make sure students understand and recall the argument.
Now suppose the players have to try and avoid connecting the opposite sides.
- Does the first player still have a winning strategy?
- Note that the strategy stealing argument doesn't apply because an extra edge can "hurt" the first player, unlike in regular Hex where an extra edge can't hurt
Game of 3 coins
There is a 1xn grid, (n>3) and there are three coins on three rightmost positions.
Each player then takes turns to move any of the coins any number of steps to the left. Coins can jump over each other but two coins can't occupy the same position.
The player who can not move loses the game.
Illustrate one game to the students, and then send them to breakout rooms to play.
- Ask the students who should win the game - first or second player. Can they prove it?
- Ask the students if they have ideas on how to win the game.
- The first player has a winning strategy, by positioning one coin on leftmost cell, and then an even number of cells gap and then the other two coins adjacent to each other.
- Therefore as long as second player can make a move, so can the first player. Additionally, the first player can always get to the invariant above.
Homework
(536Henry: Problem 297) Here is a square plot of land with four houses, four trees, a well (W) in the center, and hedges planted across the four gateways (G). Can you divide the ground so that each householder shall have an equal portion of land, one tree, one gateway, an equal length of hedge, and free access to the well without trespass?
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- Answer:
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References:
The Colossal Book of Short Puzzles and Problems, by Martin Gardner
536 Puzzles and Curious Problems, Henry Dudeney