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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: page1 page 2 page 3 Thanks for any help! (also English is not my first language so if there is anything incorrect here please let me know)

John Jackery
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  • Does this answer your question? 6 Teams Round Robin Brain Teaser – John Douma Sep 04 '21 at 08:26
  • thats also me asking the question, just made some large changes and forgot to delete it. never got an answer on it so i rephrased – John Jackery Sep 04 '21 at 08:28
  • It is expected that you show what you have tried to solve this problem. This looks like a duplicate of a question you have already asked. All of your question define context as why you need the answer but none of them show any effort on your part to solve them. – John Douma Sep 04 '21 at 08:28
  • sorry, added some png's of my working out pages – John Jackery Sep 04 '21 at 08:32
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    A couple of hints to hopefully get you unstuck: 1. Brute forcing this is not really feasible without computer assistance, you have to work backwards from the conditions you are trying to satisfy. 2. Start with a. and only move onto the other parts once you've solved it. 3. What is the range of possible numbers of home games a team can have played at any given time in the tournament? What does that tell you about how many must have played if they've each played a distinct number of home games? – hendreyth Sep 04 '21 at 09:09

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