Mathematics Magazine for Grades 1-12 Algorithm for Determining a Maximum Coupling of Minimum Value in a Bipartite Graph by Ilie Vieru Teacher- National  College Gh. Vranceanu Bacău, Romania (4)                                                                   (1), (2) , (3), (4) Exemple: In a factory, 6 workers operate on 6 machines. Let  be the losses determined by employing worker i on machine j. According to the matrix: determine the optimum order of positioning the 6 workers on the 6 machines so that the total losses are minimum.           By applying the algorithm we obtain: Step 1:        ; Step 2:        Step 3:                              Step 4:                         Step 5:                              Step 6:                                                                Therefore,       and the  order on the machines is: (1,4), (2,6), (3,5), (4,2), (5,1),  (6,3). Application: The coach’s problem (a real problem for a real coach) In a team sport ( football, rugby, hockey) where n players go on the field, the coaches of every team, together with their teams of experts, manage to determine the risk a coach is taking by deciding that player i ( ) be put to play on the position j ( ). Having this information within reach (a rectangular matrix with m lines and n columns, ), the coach has to naturally solve this problem:  to choose from the m players, the “first” n players on positions so that the total risk is minimum;  if during the game one of the players is not fit (or is injured), his replacement being necessary, to establish who of the m-n players on the bench will go on the field and what changes have to be done in the organization of the game (  modifications of positions among players) so that the total risk is also minimum. This problem, although very old, can now be solved very easily; even more, we now can rigorously explain why a coach makes some changes that the spectators “do not understand” (for example, when he replaces a forward with a mid-fielder or a defender). Biblography:  E. Tiganescu, D. Mitrut - The basis of operational research ASE Bucharest Publishing, 1994 T. Carmen - s.a. introduction in algorithms, Libris Agora Publishing Cluj Napoca, 2000 I.Vieru - The role of decision in dynamic programming, GInfo 10/3 2000. I. Tomescu - combinatorics and graph theory, Bucharest University, 1978. Read more on the written version of the publication.