6. 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?
6. 7 Let the five integers be $p, q, r, s$ and $t$ with $p \leqslant q \leqslant r \leqslant s \leqslant t$. The median of the list is $r$ and, since the mode is one less than the median, $p=q=r-1$ and $r<s<t$. The mean is one more than the median and hence the total of the five integers is $5(r+1)$. Therefore $r-1+r-1+r+s+t=5 r+5$ and hence $s+t=2 r+7$. Since the smallest possible value of $s$ is $r+1$, the maximum value of $t$ is $r+6$. Hence the largest possible value of the range of the five integers is $r+6-(r-1)=7$.