Batch 2 - Class 319 - Game of Thrones, Snails Strategy Game
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Preclass Exercise:
- (536Dudeney - 281) Two army motorcyclist, on the road at Adjbklmasdsj, wish to go to Bczrtswty, which are marked as A and B below. Pipipoff said" "I shall go to D, which is 6 miles, and then take the straight road to B, another 15 miles." But Sliponsky thought he would try the upper road by way of C. Curiously enough, they found that the distance either way was exactly the same. How far is A from C?
- Answer: 3(1/3) miles
Attendance: Anika, Yatharth, Vansh, Kabir, Aarushi, Adyant, Vivaan, Ekagra, Savya, Aneesh, Tarush, Ayush, Aditya, Rhea, Ryan, Aarav, Dhriti, Anant, Advay, Nikhil
Class Notes: (repeat from class 190)
Game of Thrones
After years of fighting and tragedy without any resolution, the leaders of Westeros decide to convene to vote among themselves for the next Ruler of the Iron Throne. Tyrion, Dany, Sansa, Cersei and Jon are the five contenders. The vote will proceed as follows: Tyrion will first nominate one of the five of them. They will all vote yea or nay for his nominee, and if that person gets at least three “yea’s”, he/she will be named the ruler. Otherwise, Dany will have an opportunity to nominate someone, and then, if that person doesn’t receive at least three votes, Sansa will choose a nominee to put up for a vote, and then Cersei will. Finally, if Cersei’s nominee doesn’t pass with three votes, John will get to appoint the next ruler (with no voting necessary).
Their preferences are as follows:
Tyrion: T > D > S > J > C (Tyrion hates Cersei.)
Dany: D > T > S > C > J
Sansa: S > C > J > T > D
Cersei: C > J > S > D > T (Cersei hates Tyrion.)
John: J > D > S > T > C
Suppose that all five of them know each other’s preferences and are strategic and forward-looking. Furthermore, suppose that they cannot commit to vote against their own preferences in the future. Finally, if indifferent between possible nominees, everyone would prefer to nominate the ultimate winner. Who will Tyrion nominate and who will win?
- Answer: D. Work backwards from John. Since Cersei knows that John will nominate himself if his turn comes, Cersai will nominate herself and win (C,S,D votes). To avoid that, Sansa will nominate herself (S,J,D votes). To avoid that, D will nominate himself (D,T,J votes). Since Tyrion will lose if he nominates himself against D, T will nominate D and let D win.
- We may start with a smaller problem. (T>S>D), (D>T>S), (S>T>D)
Now suppose that Tyrion has a chance to institute a rule that no one is allowed to nominate (or appoint, in the case of Jon) himself or herself. Would he want to institute that rule? Who would he nominate and who would win?
- Answer: Work backwards again. T will still nominate D and D will win.
Slippery Snail
Play the game below with a friend. Four bugs are placed on certain spots on the snail. Take it in turns to move any bug, moving out towards the snail's mouth (the star). You can move a bug any number of spots but bugs cannot jump on or pass each other. When a bug reaches the snail's mouth (the star), it is out of the game.
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The winner is the player who gets the last bug into the snail's mouth.
Can you find the winning strategy?
- Answer: Try to make two sets of adjacent bugs in your turn, and maintain that position
- To do that, initially bring two sets in same position (eg: distance between 1st and 2nd as 1 gap, and 3rd and 4th also as 1 gap)
- Do you see parallels to other problems? NIM games, chess rook problem
Homework:
- What if there were six bugs? Can you draw parallels to NIM games again?
- What if there are an odd number of bugs - how would you think about the problem?
References: