4. Can 64 horses be compared in 50 races to determine the superiority of any two horses? (Each race can have a maximum of 8 horses, and each horse has a different ability)
4, Solution: We can arrange the matches as follows:
(1) Divide the 64 horses into eight groups, with 8 horses in each group. Each group of horses will race once, which will rank them within the group. Thus, after 8 races, the horses in the eight groups can all be ranked within their respective groups.
(2) Combine the eight groups into four teams, with each team consisting of two groups, totaling 16 horses. Note that the best 4 horses in each team of 16 must be among the top 4 in the original two groups. Therefore, take the top 4 from each of the original two groups, totaling 8 horses to race, and the top 4 will definitely be the top 4 in the team. Remove these 4 horses from the original groups, and take the next 4 horses from each of the original two groups, totaling 8 horses to race, and the top 4 will definitely be the 5th to 8th in the team. Finally, race the remaining 8 horses in the team, which will rank all 16 horses in the team. Four teams, with a total of 12 races, will rank all the horses in the four teams.
(3) Combine the four teams into two classes, with each class consisting of two teams, totaling 32 horses. Note that the best 4 horses in each class of 32 must be among the top 4 in the original two teams. Therefore, take the top 4 from each of the original two teams, totaling 8 horses to race, and the top 4 will definitely be the top 4 in the class. Remove these 4 horses from the original teams, and take the next 4 horses from each of the original two teams, totaling 8 horses to race, and the top 4 will definitely be the 5th to 8th in the class. Continue this process, and after 6 races, 8 horses will remain. Finally, race these 8 horses, which will rank all 32 horses in the class. Two classes, with a total of 14 races, will rank all the horses in the two classes.
(4) Finally, combine the two classes into one group, which will be all 64 horses. Note that the best 4 horses in the group of 64 must be among the top 4 in the original two classes. Therefore, take the top 4 from each of the original two classes, totaling 8 horses to race, and the top 4 will definitely be the top 4 in the group. Remove these 4 horses from the original classes, and take the next 4 horses from each of the original two classes, totaling 8 horses to race, and the top 4 will definitely be the 5th to 8th in the group. Continue this process, and after 14 races, 8 horses will remain. Finally, race these 8 horses, which will rank all 64 horses in the group.
According to the above race arrangement, a total of 49 races will rank all 64 horses.