# Possible Range

The median of a set of five positive integers is one more than the mode and one less than the mean. Can you find the largest range possible?

## Problem

The median of a set of five positive integers, is one more than the mode, and one less than the mean.

What is the largest possible value of the range of the five integers?

This problem is taken from the UKMT Mathematical Challenges.

## Student Solutions

Let the median be $n$, so the numbers are:

_ , _ , $n$, _ , _

The mode is $n-1$, so the smallest two numbers must both be $n-1$, giving us:

$n-1, n-1, n,$ _ , _

The mean is $n+1$, so the total of the five numbers is $5n+5$, so the last two numbers must add up to $2n+7$.

The fourth number must be greater than $n$ (if it was $n$ there would not be a unique mode) so it must be at least $n+1$. That would give a value of $n+6$ for the fifth number.

If the fourth number was any bigger, the fifth number would have to be smaller (giving us a smaller range), so this gives the maximum range:

$n-1, n-1, n, n+1, n+6$

The difference between $n+6$ and $n-1$ is $7$,

so the maximum range is 7.