Problem 7.3. The numbers $1,2,4,5,8,10$ were placed in the cells of the figure shown in the image, such that the sums of the numbers in all columns (including the column with only one cell) were equal. What number can be in the topmost cell? List all possible options.

![](https://cdn.mathpix.com/cropped/2024_05_06_c7c94bf2f2dc4a178b54g-2.jpg?height=317&width=333&top_left_y=677&top_left_x=560)
Answer: 1, 4 or 5.

Solution. The total sum of the numbers is $1+2+4+5+8+10=30$. It follows that the sum of the numbers in each column is $\frac{30}{3}=10$; in particular, the number 10 must be in the column with only one cell.

Now, we need to notice that among the remaining five numbers, there is only one way to choose two with a sum of $10-$ this is $2+8$. They will be placed in the second column.

The remaining three numbers will be placed in the third column in any order, so the number in the top cell can be 1, 4 or 5.