Batch 3 - Class 218 - Queens, Kings and Rooks move games

• A marching band is having trouble lining up. But
• When they line up in 2's, there's 1 left over
• When they line up in 3's, there's 2 left over
• When they line up in 4's, there's 3 left over
• When they line up in 5's, there's 4 left over
• When they line up in 6's, there's 5 left over
• When they line up in 7's, there's none left over
• How many are there? Explore patterns due to 7 being relatively prime to all other numbers. What if it wasn't - suppose we extend this to 8?

• This is a 2 player game on an 8x8 chessboard. The first player places a queen on the chessboard, and the second player determines who moves first. In each move, the player can move the queen any number of steps to the left, downard, or on the left-downward diagonal. The person who makes the final move to the bottom left corner wins.
• Identify for which starting positions, would the first or the second player win
• Use a graph paper to extend and mark all squares where its better to go second

• Now instead of a queen, move a king. It means only one step in each direction. Can you create a winning strategy?
• How is the game similar to queen's move? How is it different?
• How would you describe the winning strategy in King's Move? Do you have to do it by marking squares, or can you do it more simply?

• Now instead of a queen, move a rook. It means no diagonal moves. Can you create a winning strategy?

• Find a fraction (with a whole number numerator and denominator) that is greater than 1/2020, less than 1/2019 and with the smallest possible denominator
• Answer: 2/4039 is the fraction with smallest denominator. Notice that a/b must lie between 1/2019 and 1/2020, which means that b should lie between 2019a and 2020a. b is lowest when a is lowest. a can't be 1, so the answer surfaces for a=2