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  Sage had a 1, Rosemary had an 11, and Tim had a 2. From Sage's statement, he had to have an odd number. He knew that the sum of all three numbers was even, so if his number was odd, one of the other two numbers would have to be odd, and the other even, leaving them unequal. Rosemary could deduce this, too, so when she saw an 11, she knew that Sage had a 1 and Tim had a 2. If she had seen a 12, she would have known that the other two boys each had a 1, which is contradicted by her saying that she already knew they all had different numbers. If she saw any number other than an 11 (say 9, for example), she would not have know which odd number Sage held. (In our example, he could have had a 1 and Tim a 4, or he could have had a 3 and Tim a 2). Tim, understanding all of this, therefore knew their numbers.

  If there are exactly N liars, then the last N men are truthtellers, leaving the first (12-N) men to be liars. Therefore, N=12-N, N=6. The first 6 men are liars, and the last 6 men are truthtellers.

  She wants to say, "6." The series of numbers she will say is 6, 17, 28, 39, 50. Since she wants to say 50, she needs Henry to say a number between 40 and 49. Therefore, she wants to say 39. Knowing she wants to say 39, she needs Henry to say a number between 29 and 38. So she wants to say 28. Following this same logic recursively, she will want to say 17, and she will want to say 6 to start the game, and be assured victory.

Mike is the victim. If Mike's sister and Jack's sister are the same person, then (a) reads "Lily argued exactly once with (Mike or Jack)" and (b) reads "Lily argued twice with (the same Mike or Jack)". These two statements contradict each other. Therefore Mike's sister must be different from Jack's sister. If this is the case then there are two possibilities: 1) Jack and Carol are brother and sister. Mike and Lily are brother and sister. Jack is married to Lily and Mike is married to Carol. If this is the case, then (a) reads "Carol argued once with Mike" and (b) reads "Lily argued twice with herself", leaving Jack dead. Since Lily did not likely argue with herself, this is not the solution. 2) Jack and Lily are brother and sister. Mike and Carol are brother and sister. Jack is married to Carol and Lily is married to Mike. If this is the case, then (a) reads "Lily argued once with Jack" a

  Cyclops. Why? Because the other names don't have any repeated letters

  Left: Tom, yellow, 5; Middle: Jim, red, 7; Right: Sam, blue, 6. From clue 4, the red suit's number must be greater than Sam's number, and must be odd. By trial and error, the possible combinations are (3,2), (5,4), (7,3), (7,6). From clue 1, the square can only be 324 or 576, where 3 or 7 is the red suit's number. From clue 3, the square must be 576, and 5 is the yellow suit's number and 6 the blue one. From clue 2, Tom must be the 5 and Jim the 7.