James is hosting a game show where a contestant has to draw a ball that is one of four colors from a pot. Each day, a ball of a particular color will win. To mix up the possibilities, James decides to keep unequal number of balls for each color. There will be as many black balls as red balls. Green balls and red balls should add up to 10. There should be 10 more blue balls than red balls. A math expert tells James that he should have at least 2,304 possible combinations to provide sufficient difficulty. Identify the values for the number of red balls the pot should have to ensure that James’ condition is met.

Asays "we are both knaves" and b says nothing. exercises 24–31 relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies (called normals by smullyan [sm78]) who can either lie or tell the truth. you encounter three people, a, b, and c. you know one of these people is a knight, one is a knave, and one is a spy. each of the three people knows the type of person each of other two is. for each of these situations, if possible, determine whether there is a unique solution and determine who the knave, knight, and spy are. when there is no unique solution, list all possible solutions or state that there are no solutions. 24. a says "c is the knave," b says, "a is the knight," and c says "i am the spy."