Plus Advent Calendar Door #10: Equal averages


There are several different notions of average: the mean, the median, the mode and the range (see below for the definitions). If you work out each of these for the set of numbers 2, 5, 5, 6, 7, you'll notice something interesting — they are all equal to 5!

Can you find other sets of five positive whole numbers where mean = median = mode = range?

How many sets of five positive whole numbers are there with mean = median = mode = range = 100?

This puzzle comes from our sister site Wild maths, which encourages students to explore maths beyond the classroom and is designed to nurture mathematical creativity. Visit Wild maths for more games, investigations, stories and spaces to explore!

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Given a list of $n$ numbers $x_1,$ $x_2,$ $x_3,$ up to $x_ n,$ (ordered by size) the mean is

  \[ \frac{x_1+ x_2 + ... x_ n}{n}. \]    

The median separates the lower half of the list from the higher half. It is the middle number if there are an odd number of numbers in the list, or the number half-way between the two middle numbers if there are an even number. So, assuming that $x_1 \leq x_2 \leq x_3 ... \leq x_ n.$ The median is

  \[ x_{(n+1)/2} \]    

if $n$ is odd and

  \[ \frac{x_{n/2} + x_{n/2+1}}{2} \]    

if $n$ is even.

The mode is the number in the list that occurs most often — which means that there can be more than one mode.

The range is the difference between the largest and the smallest number in the list:

  \[ x_ n - x_1. \]