Batch 2 - Class 278 - NIM Games

• Think about the solution to "how" of Divisor problem

• Describe the "Count to 20" Nim game to kids, with each player incrementing 1,2,3, or 4. Who wins the game?
• Let kids play amongst themselves for a while to discover the pattern
• Should you go first or second?
• What if you are counting to 21? Should you go first or second? For 23? 
• Who will win if we are counting to 8132? Should you go first or second?
• What if you are allowed to increment 1, 2, 3?
• What is a general way to play this game?
• Introduce notion of winning positions if kids have not already got it
• Define winning positions, and relationship with other positions (call them "losing positions")
• Generalize to deterministic strategy games
• What about counting to 100, and being able to add 1..10 in each turn?

• There are 20 pennies, with each kid able to remove 1,2,3 or 4 pennies. Player who removes the last penny wins
• See if kids can reason that this is same as Nim game
• What if we changed the rules so that the person to take the last penny loses?
• What if we can remove 1, 2 or 4 pennies at a time

• Start with a 6x8 chocolate. Break it along a line and eat one piece. Whoever eats the last piece wins
• Answer: 2x1 is a winning position to leave. For any square 3x3 or above, you can take complementary moves to leave behind a square, or if the opponent leaves less than 3 in a dimension, then leave behind a 2x1 winning position. So you should leave behind a square.

• A rook can move to left or down a 8x8 chessboard, starting at top. Player who moves last wins

• What if we had two piles of 7 and 10 counters, and each player could take one match from either or both of the two piles (Person to remove the last counter wins)