1 Aug 2013

Prize Maths Quiz: A Bus Network Puzzle (PMQ30)


An airport has a system of bus routes to shuttle passengers between terminals. The system is laid out in such a way that:

(a) there are exactly 3 bus stops on each route;

(b) it is possible to travel between any two bus stops without changing routes;

(c) any two distinct bus routes have exactly one bus stop in common.

What is the number of bus routes in the airport?

The diagram shows the fairly trivial case in which each route has just 2 bus stops; in this case, we only need 3 routes to satisfy all the conditions.

Now, there is an extension to this question. In the past, I have occasionally had a choice of two questions for a PMQ. With hindsight, I suspect this has led to some confusion, so I’m posting the extension here.



IMPORTANT CHANGES TO THE PRIZE MATHS QUIZ!

YOU MAY NOW DISCUSS THIS PMQ PROBLEM IN THE COMMENTS BELOW AND GIVE YOUR ANSWERS AND SOLUTION METHOD.

I SHALL POST A GUIDED SOLUTION ON THE FOLLOWING FRIDAY AT THE SAME TIME AS THE NEW PMQ. AS THERE ARE OFTEN MULTIPLE WAYS OF ARRIVING AT THE SAME ANSWER, I SHALL ALSO GIVE CREDIT AND LINK TO THE BEST ANSWERS SUBMITTED IN THE COMMENTS.

IF YOU ARE A STUDENT INTERESTED IN THE PRIZE BELOW, THEN YOU MAY STILL SUBMIT YOUR ANSWER BY EMAIL. BUT, FROM NOW ON, ONLY ELIGIBLE STUDENTS, PLEASE.

I HOPE THIS WILL MAKE THE COMMENTS MORE VIBRANT!

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If you would like to DONATE PRIZES for the next PMQ, send me a message here.

How to Enter

IF YOU ARE A STUDENT, you may still send your complete solution by email to pmq30markov@giftedmaths.comThis email address shall be removed after the competition closes to avoid spam. This PMQ30 competition closes on WEDNESDAY 7 AUGUST at 23:59 GMT. 

The Prize

The prizes for this PMQ are 3 free places in our Online Maths Club for ONE YEAR. The very first correct solution will receive a prize plus two others randomly selected from all the other correct answers. The email time stamp shall determine the order of entries received. All winners can have their name posted and a link to their own online profile at their favourite social network or their own blog.

Quick Rules

Look at the expanded rules on our PMQ page.
Anybody can enter our Prize Maths Quiz; adults and students.
Use the official server clock in the right column to avoid late entries.
All emails and email addresses sent to us will be deleted after the winners have been processed.
DO NOT submit your entries to the comments section at the bottom of this post.
You CAN discuss it there but you must email us to enter the competition.
We adhere to COPPA guidelines regarding children's online safety and security.
Enjoy the challenge!
Send us your solution! You can't win if you don't participate.
Then tell your friends!

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