# Task 8.3

Can the numbers $1,4,9,16,25,36,49,64,81$ be inserted into the circles in the figure so that the sum of the numbers on each radial line and in each triangle is the same?

![](https://cdn.mathpix.com/cropped/2024_05_06_7041d088f4273961b51ag-2.jpg?height=297&width=417&top_left_y=999&top_left_x=183)

Number of points 7
Answer:

impossible

Solution

Let $\mathrm{S}$ be this sum, then $6 \mathrm{~S}=2(1+4+9+16+25+36+49+64+81)$.

Since $1+4+9+16+25+36+49+64+81=285$, then $6 \mathrm{~S}=570, \mathrm{~S}=95$.

But to the sum containing the number 81, one needs to add 14 to reach 95, and there are no two numbers among them with such a sum. Therefore, the arrangement is impossible.