Question 705012

1. Understand the Total SumThe problem states that the grid is filled with numbers from 1 to 9 without repetition.

This means all 9 digits are used exactly once.

The sum of all numbers in the grid is:$S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45$

2. Analyze the Given SumsWe are provided with the sums of specific sections of the grid:

Sum of the top row = $20$Sum of the bottom row = $14$

Sum of the vertical column = $23$

If we add these three sums together, we get:

$\text{Combined Sum} = 20 + 14 + 23 = 57$3. Account for Overlapping Squares

When we calculate this combined sum, we are adding the value of every square in the grid. However, the squares where the vertical column intersects the horizontal rows are counted twice:

The top intersection square (shared by the top row and the column) contains the number $5$.

The bottom intersection square (shared by the bottom row and the column) contains the number $A$.Therefore, the combined sum equation can be written as:

$\text{Combined Sum} = (\text{Sum of all 9 numbers}) + (\text{Top intersection}) + (\text{Bottom intersection})$

4. Solve for A

Substitute the known values into the equation:

$57 = 45 + 5 + A$

$57 = 50 + A$

$A = 57 - 50$

$A = 7$