QUIZ: Fill in the blank. Necessity is the Mother of.

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1 QUIZ: Fill in the blank Necessity is the Mother of.

2 Necessity is the Mother of KLUDGE. Rube Goldberg

3 Now let s examine a fun application of binary!

4 The game(s) of Nim a.k.a. the Subtraction game German: nimm = take! Its theory was discovered by mathematics professor Charles Bouton at Harvard University in Widely played in Africa, China and Europe, at least since the Middle Ages Source:

5 The rules of Nim Rules: Two players: P1, P2 They take turns Each turn consists in removing any number of items from only one row Player who takes last item wins Source:

6 and now a little binary magic 1 stick sticks sticks sticks Source:

7 What do you notice about each 1 stick column? 3 sticks sticks sticks Source:

8 Let s define the Nim sum 1 stick sticks sticks sticks Even is zero, odd is one this is the XOR operation

9 Nim sum zero is just a fancy way of saying that all columns have even nr. of ones

10 What is the Nim sum after the last match was removed? 1 stick sticks sticks sticks Even is zero, odd is one this is the XOR operation

11 Winning strategy: If it s my move, I try to make the Nim sum equal to zero!

12 How do we make the Nim sum equal to zero on the first move? 1 stick sticks sticks sticks Even is zero, odd is one this is the XOR operation

13 How do we make the Nim sum equal Conclusion: to zero on the first move? It s impossible to make it zero (all columns even)! With perfect play, the first player always loses!!

14 It s my move - can I make the Nim sum zero? 1 stick sticks sticks sticks 0 0 1

15 It s my move - can I make the Nim sum zero? 1 stick sticks sticks sticks

16 It s my move - can I make the Nim sum zero? 1 stick sticks sticks sticks

17 It s my move - can I make the Nim sum zero? 1 stick sticks sticks 7 sticks 1 1 1

18 It s my move - can I make the Nim sum zero? 1 stick 3 sticks sticks 7 sticks Source:

19 Practice Nim! Play a few games with a friend, writing the binary numbers on a piece of paper Can you do the binary in your head? It s quite a challenge! What is the strategy if we add another row with 9 sticks? What is the strategy if the player who takes last loses? EOL 1

20 It s my move - can I make the Nim sum zero? 1 stick 3 sticks 5 sticks 7 sticks Source:

21 It s my move - can I make the Nim sum zero? 1 stick 3 sticks 5 sticks 7 sticks Source:

22 Another Nim game N = 10 Rules: Two players: P1, P2 They take turns Each turn consists in removing either 1 or 2 sticks from the one row Player who takes last wins

23 Is there a winning strategy for P1? N = 10 Each turn consists in removing either 1 or 2 sticks Player who takes last wins

24 Is there a winning strategy for P1? N = 10 Idea: Let s look at the number of sticks as a multiple of 3! 10 = 3x3 + 1 Yes, this is the remainder! In Python we write >>> 10 % 3

25 It s my move can I make the remainder zero? N = 8

26 It s my move can I make the remainder zero? N = 7

27 It s my move can I make the remainder zero? N = 9

28 It s my move can I make the remainder zero? N = 6

29 Conclusion If N is arbitrary, and I am given the option to be either the first or the second player, which one do I choose?

30 Conclusion If N is a multiple of 3, I want to be the second player If not a multiple of 3, I want to be the first

31 Another Nim game N = 10 How does the strategy change if each turn consists in removing either 1, 2, or 3 items?

32 Another Nim game N = 100 How about this game: The initial number is 100, and a player can remove any number between 1 and 12 at a time.

33 Another Nim game N = 100 The initial number is 100, and a player can remove any number between 1 and 12 at a time. Player 1 wins provided (s)he always leaves a multiple of 13 after taking: 91, 78, 65, 52, etc.

34 Food for thought N = 10 How does the strategy change if each turn consists in removing either 2, or 3 items (cannot remove 1)?

35 Assignment #3, due Wed, Sep.27 I m playing the version of Nim, and it s my move how do I make the Nim sum zero?

of Nebraska - Lincoln

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2006 The Game of Nim Dean J. Davis University of Nebraska-Lincoln