Problem 1

Consider, first, a single 2x1 ship on a 3x3 board as in the game Battleship.

  • (A) Considering all possible ways of arranging a single 2x1 ship on a 3x3 board, what are the probabilities, for each spot, that a ship lies on that spot?
  • (B) After an initial shot has been fired at the most probable spot on the board and it did not hit what are the new probabilities, for each spot, that a ship lies on that spot?
  • (C) If the second shot is fired at A1 and misses, then what are the probabilities, for each spot, that a ship lies on that spot?
  • (D) If the second shot is, instead, fired at A2 and misses, then what are the probabilities, for each spot, that a ship lies on that spot?
  • (E) Discuss which would be a better strategy, to take the second shot at A1 or at A2, based on the previous findings.

Problem 2

Consider, now, a full-sized 10x10 board.

  • (A) How many different ways are there to place a single 2x1 ship on a 10x10 board?
  • (B) How many different ways are there to place a single 3 x1 ship on a 10x10 board?
  • (C) How many different ways are there to place both a 2x1 and a 3x1 ship on a 10x10 board, following the rules of Battleship ship placement?

Problem 3

How many ways are there to place a 2x1 ship and a 4x1 ship on a 10x10 board following the rules of Battleship?

Your responses should be 1-2 pages, scanned and uploaded as a pdf file. One submission per group.