9/1/2015 HOW TO WIN ALL THE TIME (AND NOT BY CHEATING ) GAME THEORY: HOW TO WIN BMC BMC ALL THE TIME Session 1 Session 1 STRATEGIC 9.1.2015 9.1.2015 (AND NOT BY CHEATING) INVARIANTS NIM BIG IDEA: Nim is cool because it’s part of A LOT of games. WHEN YOU PLAY WITH A NEW IDEA Rules Rules: Rules Rules : : : There are many stones distributed among a bunch of squares. IN MATH, YOU WANT TO FIGURE OUT On your turn, you can pick ONE square and remove any number of stones The person to take the last stone from the game wins. HOW TO PURPOSEFULLY CHANGE THE OUTCOME. I WIN! ;D BMC Player 1: __ME__ Player 2: __ALSO ME__ THEN YOU LOOK FOR PATTERNS: Session 1 PLAYER 1 9.1.2015 WINS! WHAT CHANGES CHANGES , WHAT STAYS CHANGES CHANGES THE SAME ? WHEN SOMETHING WILL PLAYER STAYS THE SAME, IT’S CALLED 1 ALWAYS WIN THIS AN INVARIANT . . . . GAME? Goal: TWO SQUARE NIM Be the player who takes the 10-10 last stone in the whole game. 1
9/1/2015 TWO SQUARE NIM Goal: 7-7-6 Be the player who takes the last stone in the whole game. BINARY What’s going on? THIS IS BINARY: In short, we’re using a base 2 base 2 base 2 system base 2 1 = 0001 to write numbers instead of base 10. 2 = 0010 3 = 0011 Base 10 4 = 0100 3254 means 5 = 0101 “three thousands, two hundreds, 6 = 0110 five tens, and four ones” 7 = 0111 Base 8 = 1000 1101 means 9 = 1001 “one 8, one 4 10 = 1010 no 2s and one 1” 11 = 1011 AKA, it means 13 Etc. 2
9/1/2015 Nim in Binary WHAT’S THE PATTERN? NIM IN BINARY Can we find a pattern to which games are won by player 2? 0100 0010 0110 0111 0100 Step 1: Write out the number of stones in each box in binary. (= ????) 0100 Step 2: Put these numbers 0010 This is in a column and figure out 0110 called if the number of 1’s in each column is even or odd. If the 0111 it’s even, write 0 at the Nim-Sum 0100 bottom, if it’s odd write 1 0 0 1 1 1) Choose to be player… WHAT’S THE PATTERN? WHAT’S THE PATTERN? Choose to be player 1, because the Nim-sum is NON-ZERO 2) What can you do to make the Nim-Sum 0?… Find a box that has 11 at the end and take out 3 stones. 0100 0010 0110 0111 0100 0100 0010 0110 0111 0100 0100 Step 3: If the Nim-Sum Is ALL 0100 0100 ZEROS, choose to be player 2. If Nim-Sum is NOT ZERO, choose to 0010 0010 be player 1. 0110 0110 0111 0111 0100 Play: On your move, make the Play: On your move, make 0100 0100 Nim-Sum ZERO. Eventually, it the Nim-Sum ZERO. will be 0 because there are no 0 0 1 1 0 0 1 1 0000 stones left and you will win! 1) Choose to 1) Choose to be player… be player… WHAT’S THE PATTERN? WHAT’S THE PATTERN? Choose to be Choose to be player 1, because player 1, because the Nim-sum is the Nim-sum is NON-ZERO NON-ZERO 2) What can you 2) What can you do to make the do to make the Nim-Sum 0?… Nim-Sum 0?… Find a box that has Find a box that has 11 at the end and 11 at the end and take out 3 stones. take out 3 stones. Then your Then your 0100 0010 0110 0100 0100 0010 0000 0100 0100 0100 opponent moves opponent moves 0000 0010 and will and will necessarily necessarily 0100 Make the nim-sum 0100 0010 Make the nim-sum non-zero again non-zero again 0010 0010 (PROVE IT!) (PROVE IT!) 0110 0000 0000 3) REPEAT 3) REPEAT 0100 0100 STEP 2: Make the STEP 2: Make the Play: On your move, make Play: On your move, make Nim-Sum 0 Nim-Sum 0 0100 0100 Sometimes it’s the Nim-Sum ZERO. the Nim-Sum ZERO. 0 0 1 1 0 1 1 0 tricky, but you can 0110 0000 always get it back to zero again (PROVE IT!) 3
9/1/2015 1) Choose to be player… FINISH THE GAME Choose to be Considering the number of stones in each pile written in binary, player 1, because in every place value, there are an even number of 1’s the Nim-sum is NON-ZERO 2) What can you Games in which the initial Nim-Sum is non-zero. do to make the Nim-Sum 0?… Find a box that has 11 at the end and Games in which the initial Nim-Sum is 0. take out 3 stones. Then your 0010 0010 0000 0100 0100 opponent moves 0001 0001 and will necessarily 0010 Make the nim-sum non-zero again 0010 (PROVE IT!) Something that stays the same. 0000 3) REPEAT 0001 STEP 2: Make the Play: On your move, make Nim-Sum 0 0100 Sometimes it’s the Nim-Sum ZERO. 0 1 0 1 tricky, but you can A property, held by a class of mathematical objects, which remains always get it back to zero again unchanged when transformations of a certain type are applied to the objects. (PROVE IT!) FOR NEXT WEEK: CONTACT INFO My name: Zandra Vinegar My email: ch3cooh@alum.mit.edu If you want the presentation, just email me! Thanks! And have a great week! MY FAVORITE QUOTES 1) Cantor: "In mathematics the art of proposing a question must be held of higher value than solving it." 2) Howard Thurman: “Don’t ask what the world needs. Ask what makes you come alive, and go do it. Because what the world needs is people who have come alive.” 3) Richard Feynman: “Nobody ever figures out what life is all about, and it doesn't matter. Explore the world. Nearly everything is really interesting if you go into it deeply enough.” 4
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