September: Mathematics

Can you find a set of five integers which have a mean of 5, a median of 5, a mode of 5 (sets with two modes, one of which is 5 do not count), and a range of 5?

The best questions are ones where the answers lead to more questions – here are some follow up questions to consider:

  • Are there any other solutions or have you found the only one? (Can you prove that?)
  • For what values of n are there sets of n integers all of which have mean, median, mode and range of n?
  • For which values of n are there an even number of solutions? For which values of n are there an odd number of solutions? (Can you prove that?)
  • Is it possible to predict how many sets of solutions there will be given n without working them all out? Is there a neat formula?

If you are interested in studying maths then you should get as much practice of solving interesting problems as possible. Some websites you might be interested in include: www.nrich.maths.org which has an interesting selection of problems of different levels, www.mei.org.uk which has a maths item of the month and www.brilliant.org which has a lot of questions in Maths, Physics and Computer Science (some of  these are very difficult indeed but there is a system of levels that should let you get questions of the right difficulty).

Solution