1. To dig or not to dig? And where to dig? Along a straight road, there are six houses. Where should a well be dug so that the sum of the distances from it to the houses is the smallest?
1. Let points $A_{1}, A_{2} \ldots A_{6}$ be the locations of houses, and point $X$ be the location of the well. To minimize the sum $X A_{1} + X A_{6}$, point $X$ should lie on the segment $A_{1} A_{6}$. Similarly, the sums $X A_{2} + X A_{5}$ and $X A_{3} + X A_{4}$ will be minimized if $X$ lies on the segments $A_{2} A_{5}$ and $A_{3} A_{4}$, respectively. Therefore, the sum $X A_{1} + X A_{2} + X A_{5} + X A_{3} + X A_{4} + X A_{6}$ will be minimized if $X$ lies on all three segments, which means $X$ lies on $A_{3} A_{4}$. Answer: The well should be built between the third and fourth house.