Amanda drew the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-01.jpg?height=359&width=410&top_left_y=1248&top_left_x=900)

Observe that the sum along any side of the triangle above is always the same, as we can verify,

$$
1+3+6=6+2+2=1+7+2
$$

a) Complete the numbers that are missing in the circles of the figure below so that the sums along any side of the square are always the same.

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-01.jpg?height=409&width=410&top_left_y=2069&top_left_x=900)
b) Find a way to place the numbers in the circles so that the sums along any line are always the same. Is there more than one solution?

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-02.jpg?height=420&width=421&top_left_y=440&top_left_x=699)

c) In the figure below, which was only partially drawn (due to lack of space!), it is also true that the sum along each segment is always the same. However, Amanda erased all the numbers except the two numbers shown in the figure (3 and 4). It is known that there are 40 circles in the drawing. Is it possible to determine which numbers were in the circles shaded light gray and dark gray?

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-02.jpg?height=243&width=1319&top_left_y=1219&top_left_x=250)
Solution

a) In the bottom row, the sum is $2+3+5=10$. Since the sums along any side are equal, the missing number in the top right corner of the square must be equal to 2, as shown in the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-20.jpg?height=404&width=408&top_left_y=1289&top_left_x=901)

Two more numbers are missing. Again, since the sum must be 10 along any side, the missing numbers are 1 and 3, as shown in the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-20.jpg?height=407&width=404&top_left_y=1925&top_left_x=903)

b) Let's call the numbers to be placed in the top corners of the square $x$ and $y$, as shown in the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-21.jpg?height=415&width=423&top_left_y=237&top_left_x=698)

The sums must be constant along any side or diagonal drawn. The sum along the top side is $x+9+y$, so all sums must be equal to $x+9+y$. Observing the sums along the vertical sides, we deduce that the bottom corners must be equal to $y-6$ and $x+3$, as shown in the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-21.jpg?height=409&width=421&top_left_y=948&top_left_x=699)

We need to verify the sum along the drawn diagonal and the bottom horizontal side. The sum along the bottom horizontal side is equal to

$$
(y-6)+12+(x+3)=x+9+y
$$

verifying the desired sum. Verifying the sum along the drawn diagonal, we get

$$
x+17+(x+3)=x+9+y
$$

from which we conclude that $y=x+11$. Therefore, any solution will be of the form

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-21.jpg?height=410&width=420&top_left_y=1915&top_left_x=697)

Since the value of $x$ has not been fixed, we can obtain many solutions! For example, taking $x=0$, we get the solution:

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-22.jpg?height=415&width=412&top_left_y=237&top_left_x=899)

c) Let's call the numbers adjacent to the numbers 3 and 4 $x$ and $y$, as shown in the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-22.jpg?height=244&width=1319&top_left_y=843&top_left_x=443)

Since the sum along each segment is constant, the next two numbers must be equal to 3 and 4:

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-22.jpg?height=248&width=1333&top_left_y=1277&top_left_x=441)

Since there are 40 circles in the drawing, there are 20 circles in the top row and 20 circles in the bottom row. Continuing the process above, we will get:

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-22.jpg?height=249&width=1319&top_left_y=1713&top_left_x=443)

Notice the $x$ at the top right corner. This $x$ is connected to the $4$ and the $y$. Since $x+4=x+y$, we conclude that $y=4$. Therefore, the numbers in the two gray circles are equal to 4.

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