G'day, everyone. I have been asking a lot of questions today and I am very appreciative of those who have responded with advice and help and I thank you all. If you have seen my previous questions you know that the context for this question is that I had been attempting this mathematics enrichment tasks for just under 3 months now and it is finally coming to an end, however, the difficulty of the questions have increased and I am beginning to have more difficulty in answering. The final question of my 3-month journey into this booklet is less of a mathematical question and more of a brain teaser and there is no number work or pronumerals or equations, just one final brain teaser to end it all off. It is,
Six teams A, B, C, D, E and F are scheduled to play in a round robin tournament. Each team will play every other team twice, once at home and once away. For example, if teams A and B play a game at team A's stadium, it is a home game for A and an away game for B.
(a) Show that the games can be scheduled so that, at least once during the tournament, no two teams have played the same number of home games or the same number of away games.
(b) Show that the games can be scheduled so that, at least once during the tournament, no two teams have played the same total number of games.
(c) Is it possible to schedule the games so that, at least once during the tournament, no two teams have played the same number of home games, the same number of away games, or the same total number of games
The schedules for each part of the question do not need to be the same
I am looking for a way to solve this without writing down every single possibility which for each part of the question, could take hours, what are the shortcuts here and how could I apply them?
What I have attempted:
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Thanks for any help! (also English is not my first language so if there is anything incorrect here please let me know)
