22 Mar 2013

Prize Maths Quiz: A Binary Message (PMQ11)

There are a number of skills needed within mathematics competitions that rarely occur in the standard school curriculum; enumeration is one of these. Enumeration is a fancy way of saying ‘counting’, but it’s counting with style!

In secondary school, students may come across combinations and permutations as part of a course in statistics. In mathematics, these are technical terms with precise meanings and formulas. However, there are many problems that are not immediately solvable using just these types of arrangements. What is more common is that you’ll get a question that requires enumerating all possible sets that fit some criteria and then figuring out how many arrangements there are of the elements of each set. Here is one example of such a question.

The Question

A message is written with only the binary digits 1 and 0. A valid message can contain at most three consecutive 1s or 0s. For example, 11000101 and 10101010 are valid but 01111010 is not. Given that a message has a fixed length of just ten binary digits, how many valid messages can be created?

As always, I need both the final answer and a valid method.

Good Luck!

You may discuss the question below, but no full answers, please, until the competition has closed!

You can receive these questions directly to your email box or read them in an RSS reader. Subscribe using the links on the right.

How to Enter

Send your complete solution by email to pmq11newton@giftedmaths.comThis email address shall be removed after the competition closes to avoid spam. This PMQ11 competition closes on MONDAY 25 March at 23:59 GMT - one extra weekday from now on.

The Prize

The prizes for this PMQ are 3 free places in our Online Maths Club for 3 months. 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!