Three dwarfs live in different houses on a plane and walk at speeds of 1, 2, and 3 km/h, respectively. What place should they choose for their daily meetings so that the sum of the times each dwarf needs to travel from their house to this place (in a straight line) is the smallest?

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Let the desired meeting place of the gnomes be denoted by the letter $A$, and the houses be numbered $1,2,3$ according to the speeds of the gnomes. Let the distances between house 1 and houses 2 and 3 be denoted by $a$ and $b$, and the distances from point $A$ to houses $1,2,3$ be denoted by $x, y$, and $z$ respectively. Then $a / 2 \leq x / 2+y / 2$,

$b / 3 \leq x / 3+z / 3$, from which $a / 2+b / 3 \leq x / 2+y / 2+x / 3+z / 3 \leq x+y / 2+z / 3$, and equality is achieved when $x=0$. The left side is the sum of the times required for the gnomes to reach house 1, and the right side is the sum of the times required to reach point $A$, so the minimum is achieved when point $A$ coincides with house 1.

## Answer

The house of the first gnome