Definition 1 We say that straight lines in the plane are in general position if no two of them are parallel and any three of them do not intersect in one point. In figure below there are sketches of all possible configurations of three lines in the plane. Figure a. represents three lines in general position and figures b., c., and d. represent three straight lines that are not in general position. Several straight lines in the plane divide it into parts called polygonal domains. A polygonal domain can be bounded or unbounded. For example, in figure a. 6 domains are unbounded and one is bounded. How many bounded and unbounded domains are there in figures c. and d.? PROBLEM. Draw four lines in general position. Before we formulate the next problem let us imagine the following game. We have wild animals (wolves, for example) and domesticated animals (dogs). We have to place them in the plane, each animal in its own domain so that they will not be able to fight. In the bounded domains we will, of course, place wolves so they will not run away. We can place dogs in the unbounded domains because they will come back to be fed. For example, if the domains are formed by three lines in general position (see figure a.), we can place one wolf and 6 dogs. PROBLEM. Consider the domains formed by four straight lines in general position. a. How many wolves and dogs can we place on the plane? b. Do all the domains for dogs have the same shape? c. Do all the domains for wolves have the same shape? PROBLEM.^{1 } What is the largest number of bounded domains, "chambers^{2} for wolves" which can be obtained in a configuration of five lines in general position? Play with different configurations and count the number. ______________________________________________ ^{1 } This problem was the starting point for research in combinatorics published in some recent well known papers. Of course, the problem was solved for any number n of straight lines in general position. ^{2 } By the way, mathematicians also call these domains chambers but the word "wolves" is not used. |