7.3. Between the houses of four dwarfs, straight roads are laid, and no three houses are located on the same straight line. The distance between the houses of Doc and Sony is 5 km, Doc and Grumpy is 4 km, Sony and Simple is 10 km, Grumpy and Simple is 17 km. What can the distance between the houses of Sony and Grumpy be, if it is known that it is a whole number of kilometers? Provide all possible answers and prove that there are no others.

![](https://cdn.mathpix.com/cropped/2024_05_06_0f2d20123bfc334aa394g-4.jpg?height=354&width=508&top_left_y=1285&top_left_x=157)

To the solution of problem 7.3
Solution: Let's denote the houses of Doc, Sony, Grumpy, and Simple by $D, S, V$, and $P$ - see the figure. Since no three houses are collinear, the points $D, S, V$, and $P$ are the vertices of a quadrilateral (not necessarily convex). From triangle $D C V$, by the triangle inequality, we have $S V V P$, from which $S V > 17 - 10 = 7$. Therefore, $77$ or $S V7$ or |  |
| $S V$ with justification of the uniqueness of the answer) | 2 points |
| the idea of using the triangle inequality is present | 1 point |
| answer without justification (or with incorrect justification) | 0 points |
| consideration of convex and/or non-convex cases | does not change the score |