Math 111 Assignment 4 Due Thursday Nov. 3, 2016 1. Suppose that in the 7-hat game, player 1 sees a red hat on players 2, 4, 6, and 7. The other players are wearing blue hats. What should he or she do? Under what conditions will this result in a win for the group? 2. This problem concerns the code based on the 7-hat game. Here the message should be decomposed into blocks of length 7-3 = 4. Suppose that one of the 4-blocks in the starting message is 1001 (a) Find its associated L-node, which is the code word which will be transmitted. (b) Suppose that in the transmission, there is a flip in the 6th character. Show how the decoding recovers the original message. (c) Suppose that in the transmission, there is a flip in both the 1st and the 4th characters (a rare event). Then we expect that the decoding will fail to recover the message. What message is recovered? 3. Consider the 63-hat game. (a) How big is the H-matrix for this game? What is its 17th column? (b) Provide a losing configuration. (c) How many losing configurations are there? (d) Suppose there are 7 players with red hats--those whose numbers are multiples of 9. What does each of the 63 players write on his or her slip of paper? (e) Is the configuration of (d) winning? 4. This problem concerns the code based on the 31-hat game. Here the message should be decomposed into blocks of length 31-5 = 26. Suppose that one of the 26-blocks in the starting message has eight 0's followed by eleven 1's, followed by six 0's, then a 1. That is: 00000000111111111110000001 (a) Find its associated L-node, which is the code word which will be transmitted. (b) Suppose that in the transmission, there is an error in the 14th character. Give a detailed account of the calculations made by the decoder to recover the original message. math 111 assignment 4 fall 2015 1