The circles in the figure below have been filled with the numbers 1 to 7, such that all arrows point from a smaller number to a larger one. In this case, we say that the figure is well filled.

![](https://cdn.mathpix.com/cropped/2024_05_01_c051a4aaadd76ab6970bg-08.jpg?height=225&width=203&top_left_y=1989&top_left_x=1035)

a) Complete the figure below with the numbers 1 to 9 so that it is well filled.

![](https://cdn.mathpix.com/cropped/2024_05_01_c051a4aaadd76ab6970bg-08.jpg?height=243&width=308&top_left_y=2357&top_left_x=977)

b) In how many ways can the figure below be well filled with the numbers 1 to 5?

![](https://cdn.mathpix.com/cropped/2024_05_01_c051a4aaadd76ab6970bg-09.jpg?height=186&width=168&top_left_y=227&top_left_x=841)

c) In how many ways can the figure below be well filled with the numbers 1 to 7?

![](https://cdn.mathpix.com/cropped/2024_05_01_c051a4aaadd76ab6970bg-09.jpg?height=237&width=194&top_left_y=533&top_left_x=837)
Solution

a) There is only one way to fill in the diagram, as we show below.

- The number 9 cannot be placed below any number, so it must be at the top.
- Above the number 7, we can only place the 9 or the 8. Since the 9 is already at the top, the 8 will be above the 7.
- The number 6 cannot be placed below the 5 or the 2, so it will be below the 8, next to the 7.
- The number 1 is the only one that can be placed below the 2.
- The numbers 3 and 4 must be placed below the 5, with the 3 below the 4.

The sequence of figures below illustrates the steps of this reasoning.
![](https://cdn.mathpix.com/cropped/2024_05_01_c051a4aaadd76ab6970bg-34.jpg?height=532&width=1364&top_left_y=1432&top_left_x=446)

b) $1^{st}$ solution: First, let's examine the smaller diagram of three circles contained in the dotted triangle, shown below to the left. For it to be properly filled with any three distinct positive numbers, the largest number must be at the top, and the other two can be placed in the two circles below in two different ways. For example, if the numbers are 3, 6, and 8, we can arrange them in the two ways illustrated below to the right.
![](https://cdn.mathpix.com/cropped/2024_05_01_c051a4aaadd76ab6970bg-34.jpg?height=270&width=992&top_left_y=2252&top_left_x=630)

For the complete diagram of the problem to be properly filled with the numbers from 1 to 5, the 5 must be at the top. The shaded circle can be filled with any number from 1 to 4. The three remaining circles, marked with the dotted triangle, form the diagram analyzed above and can then be filled in two ways with the remaining three numbers. In summary, we can fill the diagram as follows:

- fill the top circle with the 5: one possibility;
- fill the shaded circle with 1, 2, 3, or 4: four possibilities;
- fill the three remaining circles with the remaining three digits: two possibilities.

Thus, the diagram can be filled in $1 \times 4 \times 2 = 8$ different ways. We note that this reasoning applies to any five distinct positive numbers. This will be important in solving the next item.

$2^{nd}$ solution: First, we note that the 5 must always occupy the top circle. The 4 must then occupy one of the two circles below the 5, and then:

- if the 4 occupies the shaded circle, the 3 must occupy the other circle below the 5, and the 1 and 2 can be placed in the two remaining circles in two different ways; there are two possibilities in this case;
- if the 4 occupies the other circle below the 5, the shaded circle can be occupied by any of the numbers from 1 to 3, and the other two numbers can be placed in the two remaining circles; in this case, there are $3 \times 2 = 6$ possibilities.

Thus, the total number of ways to fill the diagram is $2 + 6 = 8$.

c) $1^{st}$ solution: For the diagram to be properly filled with the numbers from 1 to 7, we must place the 7 at the top. The shaded circle can be filled with any number from 1 to 6. The part surrounded by the dotted line was analyzed in item $b$) and can be filled with the remaining 5 numbers in 8 different ways. That is, we can fill the diagram as follows:

- fill the top circle with the 7: one possibility;
- fill the shaded circle with 1, 2, 3, 4, 5, or 6: six possibilities;
- fill the part surrounded by the dotted line with the remaining digits: eight possibilities. Thus, the diagram can be filled in $1 \times 6 \times 8 = 48$ different ways.

![](https://cdn.mathpix.com/cropped/2024_05_01_c051a4aaadd76ab6970bg-35.jpg?height=363&width=325&top_left_y=1343&top_left_x=771)

$2^{nd}$ solution: First, we note that the 7 must always occupy the top circle. The 6 must then occupy one of the two circles below the 7, and then:

- if the 6 occupies the shaded circle, the numbers from 1 to 5 must occupy the circles surrounded by the dotted line. According to item $b$), this can be done in eight different ways.
- if the 6 occupies the other circle below the 7, we can place any number from 1 to 5 in the shaded circle and distribute the remaining numbers in the four still empty circles, which can be done in eight different ways, according to item $b$). Here we have $5 \times 8 = 40$ possibilities.

Thus, the diagram can be filled in $8 + 40 = 48$ different ways.