Inside each of the circles that appear in the figure below, little Luana placed one of the numbers 1, 2, 3, 4, 5, 6, and 7.

![](https://cdn.mathpix.com/cropped/2024_05_01_cc59d970f26105cc4ca1g-13.jpg?height=416&width=483&top_left_y=503&top_left_x=889)

She did this in such a way that all the numbers were used. Her older brother, Pedro, looked at each trio of collinear circles and added the three numbers placed in them. Pedro observed that the sum was always the same.

a) Show that the number placed by Luana in the top circle is 4 and that the constant sum observed by Pedro is equal to 12.

b) Find a way in which Luana could have managed to achieve this feat.
Solution

a) Let's call $s$ the sum of the numbers placed in each trio of collinear circles. Remember that this sum is always the same for each trio. Let's also call $x$ the number that Luana wrote in the top circle. The circles that are not at the top form two horizontal collinear trios. Thus, when we add the numbers placed in all these circles, we will get a result equal to twice the sum of the numbers placed in the triangles of each of the trios, that is, $2 \times s$.

The value obtained $2 \times s$, added to the number placed in the top circle $x$, must result in the sum of all the numbers placed, so

$$
2 s+x=1+2+3+4+5+6+7=28
$$

On the other hand, let's look at each trio of collinear circles that contain the top circle. Note that there are three collinear trios containing the top circle: the vertical trio and the two diagonal trios. Since the sum of the three numbers that appear in each of these trios must be equal to $s$, then the sum of the two different numbers from the number that appears at the top must be equal to $s-x$. Thus, when we add all the numbers that appear in these three trios, except the number that appears at the top, we should get a result equal to $3 \times(s-x)$. We thus have the equation:

$$
3(s-x)=1+2+3+4+5+6+7-x=28-x
$$

which is equivalent to

$$
3 s-2 x=28
$$

Equations (.7) and (.8) give us the following system:

$$
\left\{\begin{array}{l}
2 s+x=28 \\
3 s-2 x=28
\end{array}\right.
$$

Solving this system, we get that $s=12$ and $x=4$. The number placed at the top is, therefore, $x=4$.

b) We know that the number 4 must be placed at the top. Knowing that the sum must be constant and equal to 12, we can construct a way for Luana to achieve the feat:

![](https://cdn.mathpix.com/cropped/2024_05_01_cc59d970f26105cc4ca1g-39.jpg?height=357&width=400&top_left_y=233&top_left_x=931)