Pascal-like triangle of Russian Roulette @19th meeting with USTB &Tohoku University

第19回 北京科技大学-東北大学 合同研修 にて．

Transcript

1. (J-1) Tohoku University, School of Engineering, Department of Electrical, Information and Physics Engineering Yushi Nakaya Pascal-like triangle of Russian Roulette @19th meeting with USTB &Tohoku University 1
2. About this game Do you know the Russian roulette game? 2
3. About this game Russian roulette is a game of chance in which a player places a single bullet in a revolver, spins the cylinder, places the muzzle against their head and pulls the trigger. 3
4. What is a Pascal Triangle? 4
5. Definition of the terms and concepts 5 n m number of holes in the cylinder number of bullets
6. 6 Constructing the triangle for Russian Roulette number of bullets number of holes in the cylinder f(n,m) = probability player1 loses Number of players = 4 holes in the cylinder n = 5 number of bullets m = 5 1 2 2 3 2 3 1 1 1 1 1 1 1 2 1 1 3 5 3 5 4 5 1 1 2 3 3 4 7 10
7. 7 Constructing the triangle for Russian Roulette number of bullets 1 1 numerators 4 + 3 = 7 1 2 2 3 2 3 1 1 1 1 2 4 1 1 3 5 6 10 4 5 1 1 4 6 3 4 7 10 number of holes in the cylinder
8. 8 Constructing the triangle for Russian Roulette number of bullets 1 2 2 3 2 3 1 1 1 1 1 1 1 1 3 5 4 5 1 1 denominators 6 + 4 = 10 3 4 7 6 4 10 number of holes in the cylinder 2 4 6 10
9. 9 Pascal-like triangles Constructing the triangle for Russian Roulette number of bullets 1 2 2 3 2 3 1 1 1 1 1 1 1 1 3 5 4 5 1 1 3 4 4 6 7 10 number of holes in the cylinder 2 4 6 10
10. What are the conditions for making a Pascal-like triangle? 10 Question
11. 11 Purpose Find the conditions for a triangle to have a pascal-like property. Find the “if and only if condition.” (iff) 
    
12. It’s not an appropriate theme for educational research. 12 Methodology
13. How is this other game similar? 13 IUUQXXXBCJHMPCFOFKQdZPOZPODHJ"TBIJ&4""QJD@KQH empty cylinder hole with a bullet
14. There are n-m white cards and m red cards in the pile. 14 Rules of this other game All n cards are in the pile. ʜ ʜ m cards n-m cards
15. 15 ɾNumber of players = “p” each player picks up a card in turn. ɾThe player who picks up a red card loses = game over. The Reported Concept Total n cards ʜ ʜ m cards n-m cards
16. ɾPlayer A picks up s1 cards, and Player B picks up s2 cards in each turn. 16 ɾPlayer A loses when the number of red cards = g1 Player B loses when the number of red cards = g2 My new concept of this game (Russian Roulette is s1 =s2 =g1 =g2 =1)
17. 17 s1=2, s2=3, g1=2, g2=4 The pascal triangle for the card game 1 + 3 + 6 = 10 number of red cards number of cards 0 2 0 3 1 3 1 1 1 1 0 1 1 1 0 5 3 5 1 1 10 20 1 6 3 10 0 4 1 6 0 6 3 15 2 4 12 15 6 6 1 1
18. ― where the Pascal-like triangle does not hold 18 Result (1) when g1 < s1 + 1 (2) when g1 ≥ s1 + 1 (a) If g1 = 1 (b) If g1 > 1 (a) If g1 = 1 (b) If g1 > 1
19. 19 ɾOne condition was found where triangles tend to demonstrate a pascal-like property. ɾThis pascal-like condition does not always hold true. Conclusion ɾIn order to find the if and only if condition, more time and effort will be required.
20. Thank you for your kind attention. 20 Special thanks
21. 21 Message
22. The game ends here, since player B collect g2=2 cards. " " # # # " " # # # T 
    
    
    
    
    
    
    
    
    
    
    T  Example s1 =2, s2 =3, Player-A’s goal : g1 =3, Player-B’s goal : g2 =4, 22
23. 23 (1) when g 1 < s 1 + 1 ! = {$ ∈ ℕ, (, ) ∈ ℕ | + ,-. / 0, ( + 1 ≤ $ ≤ + ,-. / 0, ( + 0. } 5 = { 6 ∈ ℕ | 7. + 1 ≤ 6 ≤ + ,-. / 7, − 1 } 5 = { 6 ∈ ℕ |7. ≤ 6 ≤ + ,-. / 7, − 1} (a) if g 1 = 1 (b) if g 1 > 1 When Player A collects g1 red card, Player A will lose. Player A pics up s1 card in every term.
24. 24 (1) when g1 < s1 + 1 ! = {$ ∈ ℕ | ( )*+ , -) . + 1 ≤ $ ≤ ( )*+ , -) . + -+ ; ., 4 ∈ ℕ } 6 = { 7 ∈ ℕ | 8+ + 1 ≤ 7 ≤ ( )*+ , 8) − 1 } 6 = { 7 ∈ ℕ | 8+ ≤ 7 ≤ ( )*+ , 8) − 1} (a) if g1 = 1 (b) if g1 > 1 When Player A collects g1 red card, Player A will lose. Player A pics up s1 card in every term. If the coordinate {n,m} satisfy the condition, Pascal- like property does not hold.
25. m-coordinates n-coordinates 25 Conditions for the Triangle principle to hold �������� 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2 3 0 4 0 0 5 0 0 0 6 2 6 6 2 7 3 9 9 3 0 8 0 0 0 0 0 0 9 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 11 4 24 60 80 0 0 0 0 0 12 5 30 75 100 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 0 14 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 16 6 54 216 504 0 0 0 0 0 0 0 0 0 0 17 7 63 252 588 0 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 8 96 528 1760 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 9 108 594 1980 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
27. What is a Pascal Triangle? 27 1 1, 1 1, 2, 1 1, 3, 3, 1 1, 4, 6, 4, 1 1, 5, 10, 10, 5, 1 <latexit sha1_base64="odpDVVGJMkabGFBBjBi8veMKCNc=">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</latexit> <latexit sha1_base64="odpDVVGJMkabGFBBjBi8veMKCNc=">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</latexit> <latexit sha1_base64="odpDVVGJMkabGFBBjBi8veMKCNc=">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</latexit> <latexit sha1_base64="odpDVVGJMkabGFBBjBi8veMKCNc=">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</latexit>
29. The game ends here, since player B collect g2=2 cards. " " # # # " " # # # T 
    
    
    
    
    
    
    
    
    
    
    T  Example s1 =2, s2 =3, Player-A’s goal : g1 =3, Player-B’s goal : g2 =2, 29