Maths Solution

From Elisabeth Boghurst (Harris Girls' Academy East Dulwich)

I have found two solutions:

1. 2, 5, 5, 6, 7
Mean: 2 + 5 + 5 + 6 + 7 = 25  25/5 = 5
Median: 5
Mode: 5
Range: 7 - 2 = 5
2. 3, 4, 5, 5, 8
Mean: 3 + 4 +5 + 5 + 8 = 25  25/5 = 5
Median: 5
Mode: 5
Range: 8 - 3 = 5

I recognised that with n numbers and a mean of n they would need to total up to the square of n, in this case 25. I also put down n twice as a starting point for my median and mode. I put down two numbers with a range of n on either side and the last number (or numbers) to fit between were trial and error.

  • For what values of n are there sets of n integers all of which have mean, median, mode and range of n?

Using this method I tried to find solutions if n was less than five; I only found one for four:

2, 4, 4, 6
Mean: 4, Median: 4, Mode: 4, Range: 4

I noticed that as n had increased by one so had the number of solutions so I tried some more out of curiosity where n was greater than five:

n = 6
1. 3, 6, 6, 6, 6, 9
2. 4, 4, 6, 6, 6, 10

I didn't find three solutions but nonetheless I carried on to see if I could find any pattern . . . 

n = 7
1. 3, 5, 7, 7, 8, 9, 10
2. 4, 4, 7, 7, 7, 9, 11
3. 4, 5, 7, 7, 7, 8, 11
4. 4, 5, 6, 7, 7, 9, 11
5. 4, 6, 7, 7, 7, 7, 11
6. 5, 5, 6, 7, 7, 7, 12
n = 8
1. 4, 5, 8, 8, 8, 9, 10

This was when I had to break off and turn my attention to some homework, however I would be really interested to know if there is a pattern for the number of solutions and n, or if there is a neat formula!

Comment from James Handscombe: You have found all the possible solutions for n<=5 but there is at least one more solution for n=6. There is some symmetry in your solutions for n=5. Spotting the symmetry will help you to think about the other questions.

I don't think that there is a neat formula, unfortunately, and I expect that the number of solutions will increase enormously quickly. 

Has anyone made any more progress with this question?