Speed Problem!

Let the numbers be $a,b,c$ . Clearly $\dfrac{a+c}{2}=b$ or else the median will not equal the range. Therefore we can represent the numbers as $b-x,b,b+x$ . Now note that $2x=b$ or else the range will not equal the median. Therefore the numbers are $\dfrac{1}{2}b,b,\dfrac{3}{2}b$ . The smallest integer $b$ that makes all these values integer is clearly $b=2$ , so our numbers are $1,2,3$ and the answer is $1+2+3=\boxed{6}$ .